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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.

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.

Our aim in this paper is to show that the set $G_j(A)\cup \{0\}$, where $G_j(A)$ is the set of all J-invertible elements of a topological Jordan algebra $A$ without J-g.t.d.z, is isomorphic to $\mathbb{C}$ in the complex case. In the real case it is a $p$-normed quadratic J-division subalgebra of $A$. This result extends a well-known result of W. Żelazko ([11]) in the associative case to Jordan algebras.

In this paper, via the study of the modifications of vector bundles on the Fargues-Fontaine curve, we prove a geometric formula relating the Lubin-Tate towers with the simple basic unramified Rapoport-Zink spaces of EL type of signature $ (1, n-1), (p_1, q_1), \cdots, (p_k, q_k) $ where $ p_iq_i = 0 $. In particular, we deduce the computation of the cohomology groups of the latter., 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.

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C*-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

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.

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

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.

This paper deals with a new Hilbert- type inequality by introducing a parameter and the Beta function. As applications, the equivalent form, the reversions and some particular results are considered. All the theorems provide some new estimates on this type of inequalities.

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.

We show that the main part of the genus of the preimages of a projective line by a generic holomorphic endomorphism of CP^2 goes to the support of the Green measure., 16 pages, paper in french. Final version. To appear in Math. Ann

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 ˜.

Using the description of non solvable dynamics by Nakai, we give in this paper a new proof of the rigidity properties of some subgroups of $\operatorname{Diff}(\mathbb{C},0)$. The $\mathcal{C}^{\infty}$ case is also considered here.

We show that a sequence of smooth analytic curves of the unit ball of the complex plane, for which the genus is bounded by the area, converges to a lamination in a weak sense., 9 pages, 1 figure, paper in french

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

Our main purpose in this paper is to compute the second group of local cohomology of the current Lie algebra $\mathbb{G}_F$ with type Poincaré Lie group $G$ and stated the local deformations associated.

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