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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.

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

International audience; Let p be a prime number, and let K/k be a finite Galois extension of number fields with Galois group ∆ of order coprime to p. Let S be a finite set of non archimedean places of k including the set S_p of p-adic places, and let K_S be the maximal pro-p extension of K unramified outside S. Let G := G_S/H be a quotient of G_S :=Gal(K_S/K) on which ∆ acts trivially. Put X := H/[H, H]. In this paper, we study the ϕ-component X^ϕ of X for all Q_p-irreductible characters ϕ of ∆, and, in particular, by assuming Leopoldt conjecture we show that for all non-trivial characters ϕ, the Zp[[G]]-module X^ϕ is free if and only if the ϕ-component of the Z_p-torsion of G_S/[G_S, G_S] is trivial. We also make a numerical study of the freeness of X^ϕ in cyclic extensions K/Q of degree 3 and 4 (by using families of polynomials given by Balady, Lecacheux and more recently by Balady and Washington), but also in degree 6 dihedral extension over Q: the results we get support a recent conjecture of Gras.; Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre étranger à p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiée en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous étudions la φ-composante X φ de X pour les caractères Qp-irréductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractère non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons également une étude numérique sur la liberté de Xφ dans des extensions cycliques K{Qde degré 3 et de degré 4 (`a partir de familles de polynômes données par Balady, Lecacheux et plus récemment par Balady et Washington),et ainsi que dans des extensions diédrales de degré 6 de Q. Les résultats de nos calculs renforcent une récente conjecture de Gras

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.

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.