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 .     

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.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, a C ² submanifold, an openset with C ² boundary .Denote by (resp. ) the microlocalization along M (resp. ) of the sheaf of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups .The most general gives the vanishing outside a range of indices j whose length is equal to (with being the number of respectively positive, negative and null eigenvalues for themicrolocal Levi form ).The sharpest result gives the concentration in a single degree, provided that the difference is locally constant for near p (with for z the base point of p).The first result was restated for the complex in [D'A-Z 2], in the case codim We extend it here to any codimension and moreover we also restate for the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.     

Extremal Decomposition Problems in the Space of Riemann Surfaces

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface with a prescribed set of distinguished points to the case of the Teichmüller space corresponding to under quasiconformal homeomorphisms f. For the functional of our problem on extremal decomposition of a surface , we consider a function expressing the dependence of the extremal value of on a point . Differentiation formulas for the function are derived. These formulas are different and depend on the genus g of the surface . The case where the function is pluriharmonic is considered. Bibliography: 8 titles.     

Varieties Defined by Permutations

We continue to study interrelations between permutative varieties and the cyclic varieties defined by cycles of the form . A criterion is given determining whether a cyclic variety is interpretable in . For a permutation without fixed elements, it is stated that a set of primes for which is interpretable in in the lattice is finite. It is also proved that for distinct primes , the Helly number of a type in coincides with dimension of the dual type and equals .     

On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

Let K be a function field over finite field and let be a ring consisting of elements of K regular away from a fixed place of K. Let be a Drinfeld -module defined over an -field L. In the case where L is a finite -field, we study the characteristic polynomial of the geometric Frobenius. A formula for the sign of the constant term of in terms of leading coefficient of is given. General formula to determine signs of other coefficients of is also derived. In the case where L is a global -field of generic characteristic, we apply these formulae to compute the Dirichlet density of places where the Frobenius traces have the maximal possible degree permitted by the Riemann hypothesis.     

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 .     

Auxiliary Problem Principle and Proximal Point Methods

An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension concerns the case that the operator is split into the sum of a single-valued operator , possessing a kind of pseudo Dunn property, and a maximal monotone operator . The current auxiliary problem is k constructed by fixing at the previous iterate, whereas (or its single-valued approximation ^k) k is considered at a variable point. Using auxiliary operators of the form ^k+ , with _k>0, the standard for the auxiliary problem principle assumption of the strong convexity of the function h can be weakened exploiting mutual properties of and h. Convergence of the general scheme is analyzed and some applications are sketched briefly.     

Bernstein-Type Theorems and Uniqueness Theorems

Let be an entire function of finite type with respect to finite order and let be a subset of an open cone in a certain n-dimensional subspace (the smaller , the sparser ). We assume that this cone contains a ray 0} \right\}$$ " align="middle" border="0"> . It is shown that the radial indicator of at any point may be evaluated in terms of function values at points of the discrete subset . Moreover, if tends to zero fast enough as over , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on and , which are close to exact conditions, the function bounded on is bounded on the ray.     

Automorphisms of Aschbacher Graphs

If a regular graph of valence and diameter has vertices, then , which was proved by Moore (cf. [1]). Graphs for which this non-strict inequality turns into an equality are called Moore graphs. Such have an odd girth equal to . The simplest example of a Moore graph is furnished by a -triangle. Damerell proved that a Moore graph of valence has diameter 2. In this case , the graph is strongly regular with and , and the valence is equal to 3 (Peterson's graph), to 7 (Hoffman–Singleton's graph), or to 57. The first two graphs are of rank 3. Whether a Moore graph of valence exists is not known; yet, Aschbacher proved that the Moore graph with will not be a rank 3 graph. We call the Moore graph with the Aschbacher graph. Cameron showed that such cannot be vertex transitive. Here, we treat subgraphs of fixed points of Moore graph automorphisms and an automorphism group of the hypothetical Aschbacher graph for the case where that group contains an involution.     

Temperature-Dependence of Equilibrium States of a Two-Phase Elastic Medium with the Zero Surface Tension Coefficient

We consider the energy functional of a two-phase elastic medium with quadratic energy densities defined for such that ,where is a measurable characteristic function. Under some natural conditions on the data of the problem, we prove the existence of an interval (t ^-,t ⁺) of the change of temperature such that the energy functional has only a minimizer such that for or such that t^ + $$ " align="middle" border="0"> . The energy functional has no minimizers such that or if . We derive two-sided estimates for the numbers in terms of the characteristics of the two-phase elastic medium and the boundary condition. Bibliography: 3 titles.     

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation

The Hoff equation describes the H-beam buckling dynamics. We show that the phase space of the Hoff equation is a simple Banach manifold modeled on a subspace complementary to the kernel .     

On the Karoubi Filtration of a Category

Let be an -filtered category in the sense of Karoubi. This is the categorical analogue of an ideal in a ring . Pedersen and Weibel constructed a fibration of K-theory spectra associated with the sequence . We present a new easier proof based on Waldhausen' generic fibration.     

Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)_\mathbb{Q}

In this paper we compute the cohomology with compact supports of a Siegelthreefold as a virtual module over the product of the Galois group of over and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck--Lefschetz formula and the Arthur--Selberg trace formula.