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Abstract: We extend Saito's filtration to the Chow groups of a representable morphism of presheaves on , the category of schemes over . Moreover, for a presheaf on the category of smooth and projective schemes over , we define its higher Chow groups along with their filtrations. [Copyright &y& Elsevier]

Soit k un corps, G un groupe fini, G ↪ SLn,k un plongement. Pour k algébriquement clos, Bogomolov a donné une formule pour le groupe de Brauer non ramifié du quotient SLn,k/G. On examine ce que donne sa méthode sur un corps k quelconque (de caractéristique nulle). Par cette méthode purement algébrique, on retrouve et étend des résultats obtenus au moyen de méthodes arithmétiques par Harari et par Demarche, comme la trivialit é du groupe de Brauer non ramifié pour k = Q et G d'ordre impair. [Let k be a field, G a finite group, G ↪ SLn,k an embedding. For k an algebraically closed field, Bogomolov gave a formula for the unramified Brauer group of the quotient SLn,k/G. We develop his method over any characteristic zero field. This purely algebraic method enables us to recover and generalize results of Harari and of Demarche over number fields, such as the triviality of the unramified Brauer group for k = Q and G of odd order.] [ABSTRACT FROM AUTHOR]

We propound several problems concerning codimension one holomorphic foliations either in a local context (theory of singularities) or in a global situation (algebraic foliations). Each problem is illustrated by suitable examples. [ABSTRACT FROM AUTHOR]

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a 'rigidly' embedded invariant hypersurface. [ABSTRACT FROM AUTHOR]

Dans cet article, nous analysons en détail les deux mémoires de Legendre, Sur les intégrations par arcs d'ellipse (1786a, 1786b), en comparant les deux démonstrations qu'il donne de la rectification de l'hyperbole au moyen de deux arcs d'ellipse. Entre le premier et le second mémoire, Legendre prend connaissance d'un théorème analogue de Landen, qu'il cherche à incorporer à son projet de constitution d'un calcul des arcs d'ellipse et dont il propose une interprétation originale en le déduisant de ses propres principes. En retraçant la préhistoire de la transformation de Landen, des premières méthodes d'intégration par arcs de sections coniques chez Maclaurin, d'Alembert et Landen jusqu'à l'élaboration par Legendre des échelles de modules, nous cherchons à mettre en lumière les conditions de son identification. Nous montrons comment Legendre fut amené à discerner dans le théorème de Landen ce qui entre ses mains allait devenir l'instrument privilégié de la mise en ordre systématique des transcendantes elliptiques, grâce à une lecture orientée par les exigences spécifiques liées à une méthode d'approximation fondée sur la variation continue des paramètres et à la constitution de tables numériques d'arcs d'ellipse. [ABSTRACT FROM AUTHOR]

Abstract: The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ of a singular surface as there are essential components of a desingularisation of . It is known that this problem can be reduced to the study of quasi-rational singularities. In this Note we give a positive answer to the Nash problem for a family of non-rational quasi-rational hypersurfaces. This same method applies to give a positive answer in some other cases, for instance, the and type singularities, and gives simple proofs of known cases. [Copyright &y& Elsevier]

In this article we define and study a Harder-Narasimhan filtration of finite flat group schemes over an unequal characteristic valuation ring. We also study that type of filtrations for families of finite flat group schemes parametrised by formal schemes. [ABSTRACT FROM AUTHOR]

Abstract: We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton''s formula for the number of chains of given length in the non-crossing partitions lattice . Our proof uses the properties of the Lyashko–Looijenga covering and the geometry of the discriminant of W. [Copyright &y& Elsevier]

Soient p un nombre premier, F une extension finie de Qp, et ψ un caractère additif non trivial de F. La correspondance de Langlands donne une bijection σ ↦ π (σ) entre les représentations Φ–semisimples de dimension n du groupe de Weil–Deligne de F, à isomorphisme près, et les représentations lisses irréductibles de , à isomorphisme près. Pour certaines représentations r du groupe dual de , on sait associer, par voie globale, à une représentation lisse irréductible π de , des facteurs L(π, r, s) et ɛ (π, r, s, ψ). On conjecture l’égalité L(π (σ), r, s) = L (r ○ σ, s), et de même pour les facteurs ɛ, quand σ est une représentation de dimension n du groupe de Weil–Deligne de F. Répondant à une question de D. Jiang et D. Soudry, nous prouvons que si r = Λ2 ou , on a L(π (σ), r, s) = L (r ○ σ, s) et ɛ (π (σ), r, s, ψ) = α ɛ (r ○ σ, s, ψ), où α est une racine de l’unité. [ABSTRACT FROM PUBLISHER]

In the geometric situation of the simple Shimura varieties of Kottwitz studied in Harris and Taylor’s book (The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, ), we describe the monodromy filtration of the vanishing cycles complex and the spectral sequence associated to it. We prove in particular that this filtration coincides with the weight one up to shift. Thanks to the Berkovich-Fargues’ theorem, we deduce the description of the local monodromy filtration of the Deligne-Carayol model. [ABSTRACT FROM AUTHOR]

Berkovich (Invent. Math. 125(2):367–390, ), Fujiwara (Duke Math. J. 80(1):15–57, ) and Huber (J. Algebraic Geom. 7(2):359–403, ) have proved that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by Boyer (Invent. Math. doi:, ). [ABSTRACT FROM AUTHOR]

Abstract: We consider the Springer fiber corresponding to a nilpotent endomorphism u of nilpotent order 2. We establish a necessary and sufficient condition of singularity for the components of . To cite this article: L. Fresse, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: This article consists of two independent, but related, parts. The first one proves the vanishing of the Chow group of classes of zero-cycles of degree zero modulo rational equivalence for a cubic hypersurface of dimension ⩾10 on a p-adic or field (and, in fact, the R-triviality of such a hypersurface). This is done without the assumption of good reduction (or even smoothness). The second part goes in the other direction and gives an explicit example of a smooth cubic hypersurface of dimension 3 (necessarily of bad reduction) on a field such as (or ) whose Chow group of classes of zero-cycles of degree zero modulo rational equivalence does not vanish. [Copyright &y& Elsevier]

Abstract: We give a variant of the proof of Y.-T. Siu''s theorem concerning the non-existence of Levi-flat real hypersurface of Sobolev class , , in , . To cite this article: A. Iordan, F. Matthey, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: Let G be a reductive algebraic group over an algebraic closure of a prime field , defined over , with Frobenius endomorphism F. Let be the subgroup of rational points. Let ℓ be a prime number. Assume that ℓ is different from p. If is in duality with , then, by a theorem of M. Broué and J. Michel [M. Broué, J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989) 56–67], for any ℓ-bloc B of there exists a unique -conjugacy class of -semi-simple elements such that some irreducible representation of B is in the rational Lusztig''s series associated to . If , B is said to be unipotent. If G is not connected, with identity component , define the “unipotent ℓ-blocs of ” as the ℓ-blocks that cover some unipotent ℓ-bloc of . Assuming ℓ is good for G, we construct from and some reductive algebraic group defined over and a one-to-one map from the set of ℓ-blocks of with associated class onto the set of “unipotent”ℓ-blocks of such that, if b corresponds to B, then [–] there is a significant height preserving one-to-one map from the set of irreducible representations onto the set , [–] the respective defect groups of b and B are isomorphic, the associated Brauer categories are equivalent. One do not assume that the center of G is connected, the class of s may be isolated in . [Copyright &y& Elsevier]

Abstract: The aim of this Note is to present a very simple original, purely formal, proof of Quillen''s adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for ‘absolute’ derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. The ‘absolute derived functors’ are a non-additive generalization of Deligne''s ‘everywhere defined derived functors’. To cite this article: G. Maltsiniotis, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: We give an effective version of a result of Cohen, Shiga and Wolfart, which is a generalisation to the case of Siegel spaces of arbitrary degree, of the classical theorem of Schneider on the modular invariant . Given a point τ of the Siegel space parameterizing a principally polarised Abelian variety defined over , we obtain a lower bound for the distance between τ and algebraic points β of the Siegel space, in terms of the geometrical data of the problem. To achieve this, we establish a simultaneous measure of linear independence for periods of Abelian integrals, using Baker''s method. To cite this article: E. Villani, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: We sketch a proof of the conjecture due to Goncharov and Manin which states that the relative periods of the moduli space of Riemann spheres with n ordered marked points are multiple zeta values. To cite this article: F.C.S. Brown, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: For all , we define a p-adic regulator given by an explicit formula and we show that is non-trivial for . The main ideas come from a Note published in 2000 for the transcendental case. To cite this article: N. Hamida, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: Let Dumont has conjectured the identity , which generalizes, in particular, the classical results of Lagrange, Gauß, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont''s conjecture. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Abstract: In this Note, we give a geometric proof of Kolchin''s Theorem, using resolution of singularities. We give a counter-example in positive characteristic. To cite this article: J. Nicaise, J. Sebag, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Abstract: In this Note, we present the construction of a Neumann–Neumann domain decomposition preconditioner with scalability properties to solve nonlinear elastostatic and elastodynamic problems. By using the results given in Alart et al. [C. R. Acad. Sci. Paris, Ser. I 331 (2000) 399–404] and according to the work realized in Farhat et al. [Int. J. Numer. Methods Engrg. 38 (1995) 3831–3853] and Fragakis and Papadrakakis [Comput. Methods Appl. Mech. Engrg. 193 (2004) 4611–4662], we give the construction steps of the two level preconditioner which are based on the introduction of a coarse space especially adapted to finite deformations problems with static and dynamic process. To cite this article: M. Barboteu, C. R. Acad. Sci. Paris, Ser. I 340 (2005). [Copyright &y& Elsevier]

We give an explicit example of an Enriques surface without a rational point over C((t)). To cite this article: G. Lafon, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

Thanks to results of A. Beauville (Acta Math. 164 (1990) 211–235), we give an explicit description of translation-invariant vector fields on affine Jacobians of spectral curves, which gives a mathematical support for the work of F.A. Smirnov and V. Zeitlin (preprint math-ph/0203037). In the hyperelliptic case, this description is due to D. Mumford (Tata Lectures on Theta, Vol. II, Birkha¨user, Boston, 1983). To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]