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International audience; In my paper at Inventiones 2009, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.; Dans [2], on détermine les groupes de cohomologie des espaces de Lubin-Tate par voie globale en calculant les fibres des faisceaux de cohomologie du faisceau pervers des cyclesévanescentscyclesévanescents Ψ d'une variété de Shimura de type Kottwitz-Harris-Taylor. L'ingrédient le plus complexe consistè a contrôler lesfì eches de deux suites spectrales calculant l'une les faisceaux de cohomologie des faisceaux pervers d'Harris-Taylor, et l'autre ceux de Ψ. Dans cet article, nous contournons ces difficultés en utilisant la théorie classique des représentations du groupe mirabolique ainsi qu'un argument géométrique simple. Abstract (Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces) In [2], we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf coho-mology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

51 pages, in French, v2: extended introduction, journal version; International audience; The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

This paper is an introduction to the theory of 1d stochastic Burgers equation under periodic boundary conditions and with a stochastic force, sufficiently smooth in the space variable. We prove the classical results on the existence and uniqueness of solutions, study their regularity and discuss their properties when the time goes to infinity or the viscosity goes to zero. The latter limit describes the turbulence in the Burgers equation, named by U. Frish "the burgulence". Our paper may be used as an elementary introduction to the modern methods of stochastic PDE., Comment: in French

Dans un r\'ecent article, Lagarias et Soundararajan \'etudient les solutions friables \`a l'\'equation a+b=c. Sous l'hypoth\`ese de Riemann g\'en\'eralis\'ees aux fonctions L de Dirichlet, ils obtiennent une estimation pour le nombre de solutions pond\'er\'ees par un poids lisse et une minoration pour le nombre de solutions non pond\'er\'ees. Le but de cet article est de pr\'esenter des arguments qui permettent d'une part de pr\'eciser les termes d'erreur et d'\'etendre les domaines de validit\'e de ces estimations afin de faire le lien avec un travail de la Bret\`eche et Granville, d'autre part d'obtenir le comportement asymptotique exact du nombre de solutions non pond\'er\'ees. In a recent paper, Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we aim to prove a more precise estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bret\`eche and Granville in a recent work. We also prove the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour., Comment: 20 pages

Version incluant des corrections mineures à la version publiée. Ajout d'une note à la fin.; International audience; This paper is a continuation of the papers "Moduli spaces of decomposable morphims of sheaves and quotients by non-reductive groups" and "Espace abstraits de morphismes et mutations". I show here that the method used in the first paper to construct moduli spaces of morphisms and the one used in the second paper to extend the results of the first are particular cases of a more general method of construction of algebraic quotients. It is based on the notion of quasi-isomorphism. Let G, G' algebraic groups acting on algebraic varieties X, X' respectively. A quasi-morphism is a map $X/G\longrightarrow X'/G'$ that can be everywhere locally lifted to morphisms from open subsets of X to X' (plus some additional technical conditions...). A quasi-isomorphism is a quasi-morphism which is bijective and whose inverse is also a quasi-morphism. If there is a quasi-isomorphism between X and X'it is possible to deduce the existence of algebraic quotients of open subsets of X from the existence of quotients of the corresponding open subsets of X'. The method used to construct moduli spaces of morphisms is extended in this paper to complexes of decomposables sheaves.

Pre-print of a publication in "Annales math\'ematiques du Qu{\'e}bec". Let $k$ be a totally real number field and let $k_\infty$ be its cyclotomic $\mathbb{Z}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement r\'eels" (Annales Math\'ematiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_\infty$ ; indeed, this conjecture (on the nullity of the invariants $\lambda$ et $\mu$ of Iwasawa) depends of images in the torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, thus of Artin symbols in a finite extension $F/k$ obtained by Galois descent of ${\mathcal T}_k$. An assumption of distribution of these norms implies $\lambda=\mu=0$. Several statistics and numerical examples in the quadratic case confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg's conjecture in the sole context of Iwasawa's theory., Comment: in French, Completes with numerical computations and heuristics our previous paper arXiv:1611.09592 on Greenberg's conjecture. Annales math{\'e}matiques du Quebec, A para{\^i}tre

Let $P^+(n)$ denote the largest prime factor of the integer $n$ and $P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, firstly we show that the triple consecutive integers with the two patterns $P^+(n-1)>P^+(n)P^+(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any $J\in \mathbb{Z}, J\geqslant3$, the $J-$tuple consecutive integers with the two patterns $P^+(n+j_0)= \min\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ and $P^+(n+j_0)= \max\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ also have a positive proportion respectively. Secondly for $y=x^{\theta}$ with $0, Comment: in French

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.

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld’s p-adic symmetric space over a p-adic field K. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $$\ell $$ -adic cohomologies, as well as the torsion freeness of the integral cohomology. In this paper, we also compute the $$\ell $$ -adic cohomology of Coxeter Deligne–Lusztig variety associated to $$\mathrm{GL}_d,$$ and some results of independent interest on the coefficient systems over the Bruhat–Tits building associated to $$\mathrm{GL}_d(K)$$ have been established.

An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a+b=c$ which is valid unconditionnally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand & Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers., 31 pages, in French

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.

Euclid’s lemma can be derived from the algebraic gcd property, but it is not at all apparent that Euclid himself does this. We would be quite surprised if he didn’t use this property because he points it out early on and because we expect him to make use of the Euclidean algorithm in some significant way. In this paper, we explore the question of just how the algebraic gcd property enters into Euclid’s proof, if indeed it does. Central to Euclid’s development is the idea of four numbers being proportional: a is to b as c is to d. Euclid gives two different definitions of proportionality, one in Book VII for numbers (“Pythagorean proportionality”) and one in Book V for general magnitudes (“Eudoxean proportionality”). We will discover that it is essential to keep in mind the difference between these two definitions and that many authorities, possibly including Euclid himself, have fallen into the trap of believing that Eudoxean proportionality for numbers is easily seen to be the same as Pythagorean proportionality. Finally, we will suggest a way to make Euclid’s proof good after 2300 years.

Polarized and $G$-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group $G$ in the $G$-polarized case) and a transverse CR distribution $(E,J)$. Polarized means that $(E,J)$ is roughly speaking invariant by $\Cal F$. Both structures are therefore linked up. The interplay between them gives to polarized CR-manifolds a very rich geometry. In this paper, we study the properties of polarized and $G$-polarized manifolds, putting special emphasis on their deformations., Comment: In French. Improved statements. Main Theorems 10.1 and 13.1 are now proved without any metric condition. Also, some remarks added, typos removed and some references added

Let p be an odd prime number and K be a p-adic field. In this paper, we develop an analogue of Fontaine's theory of (phi,Gamma)-modules replacing the p-cyclotomic extension by the extension K_infty obtained by adding to K a compatible system of p^n-th roots of a fixed uniformizer pi of K. As a result, we obtain a new classification of p-adic representations of G_K = Gal(Kbar/K) by some (phi, \tau)-modules. We then make a link between the theory of (phi,tau)-modules discussed above and the so-called theory of (phi,N_nabla)$-modules developped by Kisin. As a corollary, we answer a question of Tong Liu: we prove that, if K is a finite extension of Q_p, every representation of G_K of E(u)-finite height is potentially semi-stable., Comment: 51 pages serious problem in Section 3 fixed; 2010-65

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

A proof of the transcendence of a real number ! based on the Thue‐Siegel‐Roth‐Schmidt method involves generally a sequence (" n)n! 1 of algebraic numbers of bounded degree or a sequence (xn)n! 1 of integer r-tuples. In the present paper, we show how such a proof can produce a transcendence measure for ! , if one is able to quantify the growth of the heights of the algebraic numbers " n or of the points xn. Our method rests on the quantitative Schmidt subspace theorem. We further give several applications, including to certain normal numbers and to the extremal numbers introduced by Roy.

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

This paper answers a question of Clozel and Ullmo, showing that certain sequences of adelically-defined probability measures defined on an adelic quotient of a solvable group converge to the uniform measure on that quotient. This turns out to depend on any non-trivial estimate for classical Kloosterman sums. At the end, a “horizontal” analogue of the problem is stated and solved using a result of Duke, Friedlander and Iwaniec.

In this paper we use the twistor theory in order to build non standard complex structures (with a meaning which we define) on products of 4-manifolds with the sphere of dimension two. To that end, we enumerate the set of complex surfaces whose twistor space is topologically trivial. Among these surfaces, we determine those which admit a spin structure or an anti-selfdual riemannian metric.

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.

This article completes the results of our previous paper [B.91] on interaction of consecutive strata for the vanishing cycles. In the same context that loc. cit., we treat here the tangling phenomenon for a given cohomology class of the Milnor’s fiber of f at the origin in the case where its nilpotency order for the given eigenvalue e −2iπ.u � of the monodromy is not necessarily bigger or equal than the nilpotency order of the monodromy for this eigenvalue at the generic point of the big stratum. We show that this tangling phenomenon can always be detected on the poles of the meromorphic extension of the distribution R X |f | 2λ ˜.

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.

In this paper, we associate to every $p$-adic representation $V$ a $p$-adic differential equation $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine's $(\phi,\Gamma_K)$-modules. This construction enables us to relate the theory of $(\phi,\Gamma_K)$-modules to $p$-adic Hodge theory. We explain how to construct $mathbf{D}_{mathrm{cris}}(V)$ and $\mathbf{D}_{\mathrm{st}}(V)$ from $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, which allows us to recognize semi-stable or crystalline representations; the connection is then either unipotent or trivial on $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)[1/t]$. In general, the connection has an infinite number of regular singularities, but we show that $V$ is de Rham if and only if those are apparent singularities. A structure theorem for modules over the Robba ring allows us to get rid of all singularities at once, and to obtain a ``classical'' differential equation, with a Frobenius structure. A recent theorem of Y. Andr\'e gives a complete description of the structure of such an object. This allows us to prove Fontaine's $p$-adic monodromy conjecture: every de Rham representation is potentially semi-stable. As an application, we can extend to the case of arbitrary perfect residue fields some results of Hyodo ($H^1_g=H^1_{st}$), of Perrin-Riou (the semi-stability of ordinary representations), of Colmez (absolutely crystalline representations are of finite height), and of Bloch and Kato (if the weights of $V$ are $\geq 2$, then Bloch-Kato's exponential $\exp_V$ is an isomorphism)., Comment: 71 pages. In French. Uses Xypic. 3rd Version: this revised version includes a proof of Fontaine's monodromy conjecture and some applications. Submitted for publication

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 $l(X)=l(Y)$.

Article dans revue scientifique avec comité de lecture.; Nous présentons dans ce travail une méthode, issue de travaux de Bilu et de Bilu et Hanrot, qui permet, sous certaines conditions, de borner de façon bien plus précise que par la méthode de Baker les solutions d'équations diophantiennes superelliptiques. Cette méthode s'applique en particulier à l'équation de Catalan $|x^p -y^q| = 1$, où $q > p$ sont deux premiers, et nous permet de prouver que cette dernière n'a pas de solution non triviale quand $q$ ne divise pas $h^-(\Q(\zeta_p))$. || In this paper we present a method coming from work by Bilu and Bilu and Hanrot which, under certain assumptions, allows one to obtain bounds for the solutions of superelliptic equations that are much sharper than usual bound derived through the use of

Galois theory allows us to deal with effective computation in algebraic extensions of fields. In this aim, the present paper is devoted to an inductive construction of a generating system for the ideal of relations among the roots of a univariate polynomial over a field. The idea is to define new ideals between the ideal of symmetric relations and the ideal of relations and to give a correspondence between these ideals and finite sets of permutations. The fundamental tools of this construction are multivariate polynomials called minimal polynomials associated to our ideals. These polynomials characterize the considered ideals and allow to construct a generating system for them., Cet article développe une vision effective de la théorie de Galois algébrique en apportant des propriétés inhérentes aux idéaux associés à un polynôme d’une variable. Il débouche sur un algorithme de calcul du groupe de Galois d’un polynôme et de l’idéal des relations entre ses racines.

International audience; A numeral system is a sequence of an infinite different closed normal $\lambda$-terms which has closed $\lambda$-terms for successor and zero test. A numeral system is said adequate iff it has a closed $\lambda$-term for predecessor. A storage operator for a numeral system is a closed $\lambda$-term which simulate ``call-by-value'' in the context of a ``call-by-name'' strategy. E. Tronci conjectured the following result : a numeral system is adequate if it has a storage operator. This paper gives a negative answer to this conjecture for the numeral systems typable in the J.-Y. Girard type system F. The E. Tronci's conjecture remains open in pure $\lambda$-calculus.

International audience; This paper is devoted to studying the global convergence of the Newton method generalized to systems of non differentiable equations issued from the contact mechanics. In the first part, sorne theoretical results are presented for a typical obstacle problem. In the second part, the behaviour of the Newton algorithm is analyzed by an heuristic approach. In this way, others non linearities, like friction or elastoplasticity, together with unilateral contact, can be taken up.; Cet article présente une étude de la convergence globale de la méthode de Newton généralisée aux systèmes d'équations non différentiables issues de la mécanique du contact. La première partie fournit quelques résultats théoriques pour un problème d'obstacle type. Dans la seconde partie, seule une approche heuristique permet d'appréhender le comportement de l'algorithme de Newton pour des problèmes comportant, outre le contact unilatéral, des non linéarités comme le frottement et l'élasto-plasticité.

Let k be a field of characteristic zero, F=k((Y)) the local field of Laurent series in a variable Y, and ∂ Y the usual derivation of F. The purpose of this paper is the complete description of k-derivations on the formal pseudo-differential operator skewfield K=F(XX, ∂ Y )). An application of this result is given to study the structure of higher derivations in skewfields of characteristic zero