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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

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

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 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 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

Let $n$ be an integer $\geqslant 2$. A group $G$ is called generalized $n$-abelian if it admits a {\em positive polynomial} endomorphism of degree $n$, that is if there exist $n$ elements $a_1, a_2, \dots, a_n$ of $G$ such that the function $\varphi: x\mapsto x^{a_1}x^{a_2}\cdots x^{a_n}$ is an endomorphism of $G$. In this paper we give some sufficient conditions for a generalized $n$-abelian group to be abelian. In particular, we show that every group admitting a positive polynomial monomorphism of degree 3 is abelian.

21 pages; Let $\wh K$ be the field of formal Laurent series in $X^{-1}$ over the finite field $k$, and let $A$ be the ring of polynomials in $X$ over $k$. One of the main results of the paper is to give a particularly nice coding of the geodesic flow on the quotient of the Bruhat-Tits tree $\TT$ of ${\rm PGL}_2(\wh K)$ by ${\rm PGL}_2(A)$, by using the continued fraction expansion of the endpoints of the geodesic lines in $\TT$ (the space of ends of $\TT$ identifies with $\PP_1(\wh K)$). This allows in particular to prove in a dynamical way the invariance of the Haar measure by the Artin map.

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.

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

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.

Nous r esolvons dans cet article la question de la d ecidabilit e de la th eorie monadique du second ordre du monode inversif libre. Nous d enissons a cette n la notion de th eorie monadique du second ordre d'un monode donn e et la contrepartie combinatoire de cette notion, les ensembles reconnaissables de mots g en eralis es sur un monode donn e. Nous rappelons alors la d enition du monode inversif libre ainsi que la caract erisation de ce monode due a Scheiblich. En utilisant cette caract erisation et les syst emes de pavages de Wang, nous montrons que la th eorie monadique du second ordre du monode inversif libre sur un singleton est ind ecidable, ce qui entra^ ne la propri et e pour le monode inversif libre sur un ensemble de plus d'un el ement. Abstract We solve in this paper the question of the decidability of the monadic second order theory of the free inverse monoid. To this aim, we dene the notion of monadic second order theory of a given monoid, and the combinat- oric counterpart of this notion, the recognizable sets of generalized words on a given monoid. Then we recall the denition of the free inverse monoid and the characterization of this monoid that has been given by Scheiblich. Using this characterisation and the Wang tiling systems, we show that the second order theory of the free inverse monoid on a singleton is undecidable, which entails the property for the free inverse monoid on a set which may contain more than one element.