Let D=π1..., πn, where π1, ... πn are distinct Gaussian primes ≡ (mod 4) and n is any positive integer. In this paper, we prove that the value of the Hecke L-function attached to the elliptic curve ED2: y2=x3−D2x at s=1, divided by the period ω defined below, is always divisible by 2n−1. Moreover, we give a simple combinatorial criterion for this value to be exactly divisible by 2n−1. Our results are in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer, and, when D is rational, enable us to prove the conjecture of Birch and Swinnerton-Dyer for ED2 when the value at s=1 is exactly divisible by 2n−1.

Let

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be a presentation for the group G where F = 〈a1, …, an〉 is a free group. Put A = {a1, …, an a−11, …, a−1n} and call n the degree of the presentation. For g ∈ G, c ∈ A we let W(k, g, c) be the set of freely reduced words w ∈ F having length k, which end (on the right) in c and which represent the element g ∈ G. Let w(k, g, c) = |W(k, g, c)|. The function

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will be called the cogrowth function for this presentation. Let

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Let p > 0 be prime, and let K be any field of characteristic p. Let X and Y be independent (commuting) variables over K. A polynomial f(X) ∈ K[X] is said to be additive (over K) if and only if f(X+Y) = f(X)+f(Y) in K[X, Y]. It is elementary (and classical) that the additive f(X) over K are precisely the polynomials of the type Σj=0dajXpj, where d [ges ] 0 and a0, …, ad∈K. Now let k [ges ] 2, and let X, Y, T0, …, Tk−1 be algebraically independent over K. We put q = ph (h ∈ ℕ) and define

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In this paper we present results which relate the linearization of a G-equivariant vector field at an equilibrium to the linearization of the vector field obtained by projection on the orbit space of the group action. This allows us to propose a notion of hyperbolicity for an equilibrium of the projected vector field which is compatible with the orbit space structure. It provides an alternative to the notion of normal hyperbolicity for relative equilibria (flow invariant group orbits) developed by Field [9, 10].

A countably infinite first-order structure M is ω-categorical if every countable N which has the same first-order theory as M is isomorphic to M. By a theorem due independently to Engeler[8], Ryll–Nardzewski[14] and Svenonius[19], this is equivalent to the condition that Aut M has finitely many orbits on Mn for all positive integers n. It is well known that ω-categorical algebraic structures are often intimately related to Boolean algebras. As examples, we mention Theorem 1·3, which is used later, and also the following theorem of J. S. Wilson (proved in [3]), which partly motivates our work.

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

In this paper, we introduce the notion of Jacobi polynomials for codes, and establish some fundamental features of it. This notion comes out of considerations on the various invariants of the codes (e.g. [22–24]); it has to do with Jacobi theta-series of positive definite quadratic forms (cf. [9], p. 82). Thus the name ‘Jacobi polynomials for codes’ traces this fact.

1. Introduction

A class [Xscr] of groups is said to be countably recognizable, if every group all of whose countable subgroups are contained in countable [Xscr]-subgroups is itself an [Xscr]-group. Many examples of such classes are discussed in section 8·3 of [20]. In the present work we are concerned with the question of how far countable recognizability can be obtained for classes of finitary linear groups. Recall that a group is said to be finitary []-linear if it is isomorphic to a subgroup of FGL[](V), the group of all invertible []-linear transformations α of the []-vector space V with the property that the image of the endomorphism α−idV has finite []-dimension. This generalizes the notion of linearity. A survey about features of finitary linear groups is given in [18].

The classical cobar construction ΩC for a coalgebra C (first introduced by Adams[1]) is an important algebraic concept motivated by the singular chain complex of a loop space ΩX. If X is a 1-reduced simplicial set with realization |X|, Adams proved that there is a natural isomorphism of homology groups

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where C(X) si the coalgebra given by th chain complex on X abd tge Alexander-Whitney diagonal. Here the homology has coefficients in an abelian group A. The purpose of this paper is the extension of this result to the case of twisted coefficients given by π1Ω|X|−modules A, with π1Ω|X| = H2X.

Upper bounds are established on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the Backelin–Lescot rational expression for multigraded Poincaré series.

A β-ring is supplied with operations βH where Hruns over the conjugacy classes of subgroups of the symmetric groups Sn. In an earlier paper we introduced a second set of operations λH and we show here that the two sets are related by the isomorphism

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We then consider the operations βH and λH as combinatorial species, in the sense of Joyal, and express their molecular decomposition as a finite sum of products of the exponential species with molecular species of degree at most n. We give combinatorial interpretations for βSn-structures and λSn-structures and derive various species isomorphisms.

Let V be a 4-dimensional complex space. A congruence Y is an integral surface of the Grassmann variety G = Gr(2, 4) of 2-dimensional subspaces V2 of V (we denote by Vi a subspace of V of dimension i). They have been extensively studied by both classical and modern geometers. We bring to their study the tool of Chow forms, characterizing them by differential equations, following the program of M. Green and I. Morrison [3]. The first results in this direction are due to Cayley ([1], [2]) and are rederived in [3]. Our results share much of the geometrical flavour of Cayley's.

We study algebraic models of the Segal category of Riemann surfaces [11]. These are 2-categories based on the 1+1 dimensional cobordism category enriched by diffeomorphisms of surfaces (or isotopy classes of such). In particular, we show that one of these models is a symmetric strict monoidal strict 2-category. The main technical tools are quotient constructions on 2-categories.

For E a subset of ℝn and 0 [les ] m [les ] n we define a ‘family of dimensions’ DimmE, closely related to the packing dimension of E, with the property that the orthogonal projection of E onto almost all m-dimensional subspaces has packing dimension DimmE. In particular the packing dimension of almost all such projections must be equal. We obtain similar results for the packing dimension of the projections of measures. We are led to think of DimmE for m ∈ [0, n] as a ‘dimension profile’ that reflects a variety of geometrical properties of E, and we characterize the dimension profiles that are obtainable in this way.

Recently, Hitoshi Murakami has shown that the quantum SU(2)- and SO(3)-invariants of 3-manifolds at roots of unity of prime order are algebraic integers. Unfortunately, his proof is by a very complicated computation. Here, a quite different and very simple proof is presented, based on the second author's method to show that the Turaev–Viro invariant is the square of the modulus of the Reshetikhin–Turaev invariant.

1. Introduction

It has been known for some time that contact structures show a high degree of topological flexibility in the sense that many topological operations can be performed on contact manifolds while preserving the contact property. For instance, Martinet [14] used a surgery description of 3-manifolds to show that every closed, oriented 3-manifold admits a contact structure, and alternative proofs of this result were given later by Thurston and Winkelnkemper [18], who based their proof on an open book decomposition, and Gonzalo [8], who used branched covers. These, however, are all strictly 3-dimensional constructions.

For a Radon probability measure μ on ℝn we can use the Hausdorff dimension dimH and the packing dimension dimp to define lower indices

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and

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For two such measures μ and ν intersection measures μ ∩ f# ν are obtained by slicing a certain product measure on ℝn × ℝn using diagonal n-planes, and by projecting the result onto ℝn. Here f is a similarity on ℝn and f# ν is the image of the measure ν under f. Our main result gives a lower bound for the packing dimension of these intersection measures under the condition dimH μ+dimH ν > n and some additional assumptions.

In this paper we give for any r, n, 0 [les ] r [les ] n, a Quillen's model structure to the category of simplicial groups where the weak equivalences are those morphisms f[bull] such that πq(f[bull]) is an isomorphism for r [les ] q [les ] n. This is carried out by studying the cases r = 0 and n → ∞ previously and, in each one of them, we make explicit some constructions for the associated homotopy theories, such as the cylinder and path objects and the loop and suspension functors, and we also relate the simplicial homotopy relation to the homotopy relation obtained from these structures.

Real-valued function spaces CoL lack almost transitive norms. If there is a complex-valued CoL with almost transitive norm then there is also one with transitive norm. We give fairly restrictive topological conditions for the complex-valued CoL to have transitive norm.

Given an analytic path of flat connections on a principal U(N)-bundle over a closed odd-dimensional manifold M, how can we calculate the spectral flow of the corresponding path Dt of signature operators on an associated vector bundle E → M? In this paper we give an algebraic-topological answer to this question in terms of cohomology, cup products and higher Massey products.