Rang des courbes elliptiques et sommes d'exponentielles

Lety ²=x ³+a(t)x+b(t) be the equation of an elliptic curve over , wherea(t) andb(t) are polynomials over. We give an upper bound on average of the rank of such curves whent varies over under the assumption that theL-function attached to these curves satisfies classical assumptions. The proof is based on estimates of exponential sums, over varieties, which are treated by Deligne's theorem.

     

Sur la caractéristique d'Euler des feuilletages mesurés

On prouve que la caractéristique d'Euler d'un feuilletage mesuré est nulle si et seulement si celui-ci admet un champ tangent transversalement mesurable dont l'ensemble de singularités est de mesure arbitrairement petite. Si le feuilletage est moyennable, alors on peut construire un tel champ sans singularités. Si les feuilles sont de dimension deux, l'annulation de la caractéristique d'Euler implique la moyennabilité ; plus encore, elle équivaut à l'existence d'une action mesurable de R² sur la variété ambiant qui est continue et localement libre sur chaque feuille.     

The <Emphasis Type="Italic">L</Emphasis>-series of a cubic fourfold

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface S, defined over a number field K and satisfying , then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in terms of the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example.     

Convergence des EDSRs et homogéneisation des inégalités variationnelles semilinéaires dans un convexe

We study the limit of the solution of a Semi-linear Variational Inequality (SVI for short) involving a second order differential operator of parabolic type with periodic coefficients and highly oscillating term. Our basic tool is the approach given by Pardoux [16]. In particular, we use the weak convergence of an associated reflected Backward Stochastic Differential Equation (BSDE for short).     

Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Let be a hyperelliptic curve of genus over a discrete valuation field . In this article we study the models of over the ring of integers of . To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of over dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.

     

Moduli spaces of principal <Emphasis Type="Italic">F</Emphasis>-bundles

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation .of of Our spaces generalize moduli spaces of F-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case G = GL^r and is the tensor product of the standard representation and its dual. The importance of the moduli spaces of F-bundles is due to the belief that Langlands correspondence is realized in their cohomology.     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

The Néron model over the Igusa curves

We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3, where wild ramification and stacky phenomena show up.     

Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">d</Emphasis></Subscript>

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D _d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when and successively we extend the result to curves of genus g ≥ 1.     

On Shimura subvarieties generated by families of abelian covers of P1

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of ${\mathbb{P}}^1$ in $A_g$ and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus $T_g$ inside $A_g$. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.     

Valuations,intersections et fonctions de Belyi-quelques remarques sur la construction des hauteurs

We present in this article several possibilities to approach the height of an algebraic curve defined over a number field: as an intersection number via the Arakelov theory, as a limit point of the heights of its algebraic points and, finally, using the minimal degree of Belyi functions.     

Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.     

Le problème de Yamabe avec singularités

Let (Mn,g) be a compact riemannian manifold of dimension n?3. Under some assumptions, we prove that there exists a positive function φ solution of the Yamabe equation

     

On thec-spectral sequence for systems of evolution equations

The groups E ₁ ^2,n–1 ( ) of Vinogradov'sC-spectral sequence for determined systems of evolution equations are considered. Presentation of these groups useful in practical computations is obtained. The group E ₁ ^2,1 ( ) is calculated for a system of Schrödinger type equations.