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Dans cet article, nous nous intéressons aux dimensions de certaines variétés qui ont été introduites récemment par Kisin pour démontrer la modularité de certaines représentations galoisiennes. Nous étudions plus spécialement un cas particulier pour lequel nous donnons une estimation de la dimension en question, puis, en nous basant sur ce résultat, nous énonçons une conjecture dans le cas général. In this paper, we study dimensions of some varieties, that were introduced recently by Kisin in order to prove modularity of some Galois representations. In fact, we mainly consider a special case for which we obtain an estimation of the dimension we are interested in. Then, based on this result, we state a conjecture for the general case.

In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of a radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of $2 \times 2$ matrices whose coefficients are the so called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the Complex Angular Momentum method and recent results on 1D inverse problem from generalized Weyl-Titchmarsh functions, we show that the knowledge of the reflection operators at a fixed non zero energy is enough to determine uniquely the metric of the asymptotically hyperbolic Liouville surface under consideration., 39 pp

Let B ( ϑ ) B (\vartheta ) denote the first normalised Bernoulli function, and consider the series f ( ϑ ) := ∑ n ⩾ 1 B ( n ϑ ) / n f(\vartheta ):=\sum _{n\geqslant 1}B(n\vartheta )/n . In a previous paper, we determined the set E ∗ E^* of those real numbers ϑ \vartheta at which f ( ϑ ) f(\vartheta ) converges. Let B 2 ( ϑ ) B_2(\vartheta ) designate the second Bernoulli function and let ⟨ t ⟩ \langle t\rangle denote the fractional part of the real number t t . Put F ( ϑ ) := ∑ n ⩾ 1 B 2 ( n ϑ ) / 2 n 2 = π 2 / 72 + ∫ 0 ϑ f ( t ) d t . F(\vartheta ):=\sum _{n\geqslant 1}{B_2(n\vartheta )/ 2n^2}=\pi ^2/72+\int _0^\vartheta f(t)\mathrm {d} t. It has been shown by Báez-Duarte, Balazard, Landreau and Saias (2005) that F ( ϑ ) F(\vartheta ) and ∫ 0 ∞ ⟨ t ⟩ ⟨ ϑ t ⟩ d t / t 2 \int _0^\infty \langle t\rangle \langle \vartheta t\rangle \mathrm {d} t/t^2 have the same differentiability points. Moreover, using delicate functional equations, Balazard and Martin recently proved that the corresponding set is precisely E := E ∗ ∖ Q E:=E^*\smallsetminus \mathbb {Q} and that F ′ ( ϑ ) = f ( ϑ ) F’(\vartheta )=f(\vartheta ) whenever ϑ ∈ E \vartheta \in E . We provide a short, direct proof of this last result, based on standard results from Diophantine approximation theory and uniform distribution theory.

In this short paper, we will provide a characterisation of interpretable groups in a beautiful pair (K, E) of algebraically closed fields : every interpretable group is, up to isogeny, the extension of the subgroup of E-rational points of an algebraic group by an interpretable group which is the quotient of an algebraic group by the E-rational points of an algebraic subgroup.---Dans une belle paire (K;E) de corps alg\'ebriquement clos, un groupe d\'efinissable se projette, \`a isog\'enie pr\`es, sur les points E-rationnels d'un groupe alg\'ebrique ayant pour noyau un groupe alg\'ebrique. Un groupe interpr\'etable est, \`a isog\'enie pr\`es, l'extension des points E-rationnels d'un groupe alg\'ebrique par un groupe interpr\'etable, qui est lui le quotient d'un groupe alg\'ebrique par les points E-rationnels d'un sous-groupe alg\'ebrique., Comment: in French

Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A) the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL_1(A) the algebraic k-group whose points are invertible elements of A. The group GL_1(A) acts naturally on Gr(d,A) (by the formula g.E=gE). The aim of this paper is to study some birational properties of this action. More precisely, let r be the gcd of d and dim(A). Under some hypothesis on A (satisfied if A/k is \'etale), I show that the variety Gr(d,A) is birationally and GL_1(A)-equivariantly isomorphic to the product of Gr(r,A) by a projective space (on which GL_1(A) acts trivially). By twisting, this result has some corollaries in the theory of central simple algebras. For instance, let B and C be two central simple algebras over k, of coprime degrees. Then the Severi-Brauer variety SB(B \otimes C) is birational to the product of SB(B) \times SB(C) by an affine space of the correct dimension. These corollaries are in the spirit of Krashen's generalized version of Amitsur's conjecture., Comment: 15 pages

Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n)., 50 pages, french. Section 4.18 has been extended : let F be a totally real field and \pi a selfdual cuspidal automorphic representation of GL(2n,A_F) which is cohomological at all the archimedean places and discrete at a finite place at least, we show that for each place v the L-parameter of \pi_v preserves a non-degenerate symplectic pairing

23 pages, in French; The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety -- provided this coherent sheaf is a subquotient of a hermitian vector bundle -- and using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of the so constructed arithmetic Hilbert-Samuel function.

International audience; This paper analyses a nonlinear age-maturity structured system which arises as a model of the blood cellular production in the bone marrow. The resulting model is a nonlinear first-order partial differential equation in which there is a distributed temporal delay and a retardation in the maturation variable. We prove that uniqueness of solutions depends only on small maturity cells (stem cells) and we give a result of invariance.

International audience; An integer n is called friable when its prime number factorization consists exclusively of (relatively) small factors. Let P(n) be the largest prime factor occurring in this factorization, and let S(x,y) denote the set {n? x,P(n)? y}. This work investigates the asymptotic behaviour of the summatory function Psi_f(x, y):=\sum_{n\in S(x,y)}f(n) when f is a multiplicative arithmetical function satisfying some simple and general conditions concerning its mean behaviour on primes and on powers of primes. One such condition is |\sum_{p? z}f(p)\log p-\kappa z|? Cz/R(z) (z>1), where C is a constant and \kappa>0; requirements on R are technical conditions too long to state here, but satisfied by any "reasonable" positive increasing function. By setting R(z)=(\log z)^\delta in the very general Théorème 2.1, the authors obtain a more general as well as more precise estimate on \Psi_f(x, y) (Corollaire 2.2) than that recently obtained by J. M. Song [Acta Arith. 102 (2002), no. 2, 105--129; MR1889623 (2003a:11123)]. Their next result (Corollaire 2.3) offers a general estimate in the case where f(p) is on average very close to a constant, and contains without loss of precision estimates of the literature for particular functions f, such as the so-called Piltz functions \tau_k, or the function µ^2 where µ is the Möbius function. Then, as a further application of their first result, they establish an Erdös-Wintner theorem on friable integers (Théorème 2.4). They finally mention an application to the case where f(n) is the characteristic function of the integers that can be represented as a sum of two squares of integers (Théorème 2.5); their estimate is uniformly valid for x ? 3, exp((\log x)^{2/5+\epsilon})? y? x. The paper begins with an historical introduction, and is followed by an extensive bibliography on the subject.

National audience; In this paper we study the asymptotic limit of the solutions of Landau-Lifschitz equations as the exchange coefficient goes to zero. In order to perform this limit in regular spaces, we have to take into account the singularity induced by a boundary layer term.; On étudie le comportement asymptotique des solutions des équations de Landau-Lifschitz lorsque le coefficient d'échange tend vers zéro. Une couche limite se forme alors au bord du matériau, et on décrit cette couche limite.

International audience; The Lindelöf number $l(X)$ of a Tychonoff space $X$ is the smallest infinite cardinal $\tau$ such that any open cover of $X$ contains a subcover of cardinality less than or equal to $\tau$. The symbol $C_p(X)$ denotes the space of real-valued continuous functions on $X$ endowed with the topology of simple convergence. A well known fact is that if $C_p(X)$ and $C_p(Y)$ are isomorphic as topological rings, then $X$ and $Y$ are homeomorphic. The main resul of this paper shows that if $C_p(X)$ and $C_p(Y)$ are linearly homeomorphic, then $l(X)=l(Y)$.