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 .     

Invariants Cohomologiques de Rost En Caractéristique PositiveRost''s Cohomological Invariants in Positive Characteristic

Let G/F be a semisimple algebraic group defined over a field F with characteristic . Let us denote by the Galois cohomology group introduced by Kato. If , we show that the p-primary part of Rost's invariant lifts in characteristic 0. This result allows to deduce properties of the Rost invariant in positive characteristic from known properties in characteristic 0. The case of Merkurjev–Suslin's invariant is specially interesting, i.e. if G/F=SL(D) for a central simple algebra D/F with degree p and class , one has and an element is a reduced norm if and only if the cup-product is trivial in ; one characterizes also in positive characteristic fields with p-dimension by the surjectivity of reduced norms.In a second part, we study Rost invariants when the base field is complete for a discrete valuation. As planned by Serre, invariants are then linked with Bruhat–Tits' theory, this yields a new proof of their nontriviality.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

On the Hellinger type distances for filtered experiments

Summary We study the Hellinger type distances on a filtered space. Herep2 is an arbitrary number andP _T and are two probability measures stopped at a random timeT. We give lower and upper bounds for in predictable terms.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Realization and Smoothness Related to the Laplacian

For f L _n (T ^d ) and , the modulus of smoothness

Regularity of a Boundary Point for Degenerate Parabolic Equations with Measurable Coefficients

We investigate the continuity of solutions of quasilinear parabolic equations in the neighborhood of the nonsmooth boundary of a cylindrical domain. As a special case, one can consider the equation with the p-Laplace operator p. We prove a sufficient condition for the regularity of a boundary point in terms of C _p-capacity.     

C*-Modular Vector States

Let be a C*-algebra and X a Hilbert C* -module. If is a projection, let be the p-sphere of X. For φ a state of with support p in and consider the modular vector state φ_x of given by The spheres provide fibrations

On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –_p is the p-Laplace operator, p > 1 and is a C ^1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.     

On regularities of the distribution of special sequences

Ifp2 is an integer, then every nonnegative integerk is represented by an expression of the form with integersa _i (k), 0a _i (k)p–1,i=0.1,...,s. The radical-inverse function to the basep, _p (k), is defined by . The sequence is uniformly distributed modulo 1 (it may be called a one-dimensional Halton sequence). In the casep=2 it is the van der Corput sequence. The set of all numbers (0, 1] such that the local discrepancy is bounded inn is determined.     

On the Cyclotomic Dedekind Embedding and the Cyclic Wedderburn Embedding

Let n1 and let p be a prime. Expand j[0,p ⁿ –1]\(p) p-adically as j= _s0 a _s p ^s with a _s [0,p–1]. The #([0,j]\(p))th Z _(p)[ _p ⁿ ]-linear elementary divisor of the cyclotomic Dedekind embedding

On the best polynomial approximation of entire transcendental functions in Banach spaces. II

We study the behavior of the best approximations of entire transcendental functionsf(z) of the order =0 by polynomials of at mostn th degree in the metric of the spaceE _p(),p1. In particular, we describe the relationship between the best approximations and the logarithmic order _L and type _L of the functionf(z).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1318–1322, October, 1994.     

Gamma limits in some Bernoulli free boundary problem

We study the limit cases p   and p  1 of the functionals

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Stickelberger Codes

Let p be an odd prime and be a primitive p th root of unity over . The Galois group G of over is a cyclic group of order p-1. The integral group ring [G] contains the Stickelberger ideal S _p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S _p (q) obtained as reductions of S _p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S _p (p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B _2k , , which are divisible by p. We then develop methods to compute the generator polynomial of S _p (p). This gives rise to anew algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.     

Regularity of a boundary point for singular parabolic equations with measurable coefficients

We investigate the continuity of solutions of quasilinear parabolic equations near the nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for the regularity of a boundary point, which coincides with the Wiener condition for the Laplace p-operator. The model case of the equations considered is the equation with the Laplace p-operator _p for 2n / (n + 1) < p < 2.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 506–516, April, 2004.     

An extremal problem for empirical measures under dependent Gaussian observations

One describes a class of metricsp in the space of probability distributions on the line, for which the minimum of the mean value of the random variablep(F _x ^* , F _y ^* ), where X, Y are independent random variables, distributed according to the Gauss law , is attained at =I.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 164–166, 1985.The author is grateful to B. S. Tsirel'son for stimulating discussions.     

Sur la construction d'un corps de Hilbert

We are concerned with the following: If k is a quadratic field and N a cyclic unramified extension of degree qⁿ over k, q a prime number, determine N explicitely via a primitive element , i.e., N=k(), in the spirit of Helmut Hasse [3]. We propose a method which determines these extensions, once we are able to specify the arithmetic of a certain field . To explicit our method, we construct the Hilbert fields of (226) and (646).     

Bergman Kernels for Almost Spherical Domains with Boundaries of the Class H σ

This paper is devoted to the behavior of the Bergman kernels for almost spherical strictly pseudoconvex domains in . We find several first terms of the asymptotics for the Bergman kernel of a strictly pseudoconvex domain with H -smooth boundary, 4 < < 6. New terms in comparison with the case of C ⁶-smooth boundary appear but the growth rate of the remainder remains the same. Bibliography: 16 titles.