Sur quelques représentations modulaires et p-adiques de GL2(Q p ): I

Let p be a prime number and F a complete local field with residue field of characteristic p. In 1993, Barthel and Livné proved the existence of a new kind of -representations of GL₂(F) that they called 'supersingular' and on which one knows almost nothing. In this article, we determine all the supersingular representations of GL₂(Q _p ) with their intertwinings. This classification shows a natural bijection between the set of isomorphism classes of supersingular representations of GL₂(Q _p ) and the set of isomorphism classes of two-dimensional irreducible -representations of .     

Twists of elliptic curves

If E is an elliptic curve over , then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes for which has positive rank. For some special curvesE we show that there is a set S of primes p with density for which if is a squarefree integer where , then E(D) has rank 0. In particular E(p) has rank 0 for every . As an example let E1 denote the curve .Then its associated set of primes S₁ consists of the prime11 and the primes p for which the order of the reduction ofX₀(11) modulo p is odd. To obtain the general result we show for primes that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities.     

Graded Jacobi operators on the algebra of differential forms

One-to-one correspondences are established between the set ofall nondegenerate graded Jacobi operators of degree -1 defined onthe graded algebra of differential forms on a smooth, oriented,Riemannian manifold M, the space of bundle isomorphisms , and the space of nondegenerate derivations of degree 1 havingnull square. Derivations with this property, andJacobi structures of odd -degree are also studied throughthe action of the automorphism group of .     

Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

For a smooth projective variety X of dimension n in a projective space defined over an algebraically closed field k, the Gauss mapis a morphism from X to the Grassmannian of n-plans in sending to the embedded tangent space .The purpose of this paper is to prove the generic injectivity of Gauss mapsin positive characteristic for two cases; (1) weighted complete intersectionsof dimension of general type; (2) surfaces or 3-folds with -semistable tangent bundles; based on a criterion of Kaji by looking atthe stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman--Piene is true in case X is of general type of dimension . The second result is a generalization of the injectivity for curves.     

q-Pseudoconvexity and Regularity at the Boundary for Solutions of the \bar \partial -problem

For a domain of we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the -complex for forms with -coefficients in degree . All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the L ²-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-L ² and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

Quasisimilar Weak Contractions Have Isomorphic Lattices of Invariant Subspaces

A contraction T acting on a Hilbert space H is called a weak contraction if the spectrum of T does not cover the unit disk and the operator I-T ^* T is of trace class. Operators T₁:H₁ H₁ and T₂:H₂ H₂ are called quasisimilar if there exist operators >X:H₁ H₂ and Y:H₂ H₁ such that T₂X=XT₁, YT₂=T₁Y, and X and Y have zero kernels and dense ranges. It is proved that if two weak contractions T₁ and T₂ acting on separable spaces H₁ and H₂ are quasisimilar, then there exists an operator X:H₁ H₂ such that XT₁=T₂X and the mapping , where E=clos XE for E Lat T₁, is a lattice isomorphism. An example is given of two quasisimilar weak contractions such that for any isomorphism , its inverse is not equal to for a (bounded) operator Y. Bibliography: 4 titles.     

Moduli Spaces of Stable Polygons and Symplectic Structures on\overline {\mathcal{M}} _{0,n}

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to , brings up a new tool to study the Kähler topology of . A wild but precise conjecture on the shape of the Kähler cone of is given in the end.     

Levi Classes Generated by Nilpotent Groups

Let be a class of all groups G for which the normal closure (x) ^G of every element x belongs to a class . is a Levi class generated by . Let and ₀ be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that and , and so and . It is shown that quasivarieties and are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that is closed under free products if so is .     

The de Rham Bundle on a Compactification of Moduli Space of Abelian Varieties

In this paper, we show that the Chern classes c _k of the de Rham bundle defined on any good toroidal compactification of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of , for g=4, 5 and k>0.     

Brauer Diagrams, Updown Tableaux and Nilpotent Matrices

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg's result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all where is a complete flag in is a nondegenerate alternating bilinear form on and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both and instead of Steinberg's variety of where are two complete flags in and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both .     

On the Group of Automorphisms of Shimura Curves and Applications

Let V _D be the Shimura curve over attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of V _D and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over and over and we use it to study the set of rational points on V _D over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of V _D .     

On the Asymptotic Behaviour of General Partition Functions,II

Let = {a ₁, a ₂,...} be a set of positive integers and let p (n) and q (n) denote the number of partitions of n into a's, resp. distinct a's. In an earlier paper the authors studied large values of log(max (2,p (n)))/log(max(2,q (n))). In this paper the small values of the same quotient are studied.     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.