The main aim of this paper is to present a general method of proving Ω-estimates for a class of arithmetic error terms. We assume that error terms in question are boundary values of harmonic functions on the upper half-plane satisfying certain subsidiary conditions. We prove a general theorem for an axiomatically defined class of such functions and then we show how this result can be used to give statements in concrete situations. As examples we treat the classical case of the remainder term in the prime number formula obtaining a new proof of the well-known result of J. E. Littlewood, and the case of the remainder term in the asymptotic formula for the summatory function of the square-free divisor function. In the latter case our result is new.

Let E be an elliptic curve over , and τ an Artin representation over that factors through the non-abelian extension , where p is an odd prime and n, m are positive integers. We show that L(E,τ,1), the special value at s=1 of the L-function of the twist of E by τ, divided by the classical transcendental period Ω+d+|Ω−d−|ε(τ) is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.

Within the Tate–Shafarevich group of an elliptic curve E defined over a number field K, there is a canonical subgroup defined by imposing stronger conditions at the places above a given prime p. This group appears naturally in the Iwasawa theory for E. We propose a study of what one can say about the relation to the full Tate–Shafarevich group. Some numerical examples are included, as well as a few conjectures.

The Brauer–Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer–Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many primes p; this is a process which may be time-consuming. The result described in this paper shows that, at some primes, we do not need to list all p-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulo p.

In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

In their 1990 paper, Bloch and Kato described the image of the Kummer map on an abelian variety over a local field, as the group of 1-cocycles which trivialise after tensoring by Fontaine's mysterious ring BdR. We prove the analogue of this statement for the universal nearly-ordinary Galois representation. The proof uses a generalisation of the Tate local pairing to representations over affinoid K-algebras.

In this paper we describe the 2-category 2Mat of 2-matrices, i.e., a strict totally coordinatized version of the 2-category 2Vect of Kapranov and Voevodsky finite dimensional 2-vector spaces, analogous to the category Mat of matrices which appears as the coordinatized version of the category Vect of finite dimensional vector spaces. In particular, explicit formulas for the composition of 1-morphisms and the two compositions between 2-morphisms are given.

For α∈ (0,1) and x∈ [0,π, we define S0(x,α) = 1 and where Also, we set, for α∈ (0,1), where it is agreed that infØ =∞. It is shown in this paper that and 8≤ N(α)<∞ if α0<α<α*, where α*:=0.33542. . . is the unique solution α∈ (0,1) of the equation while α0= 0.308443. . . is the unique solution α∈ (0,1) of the equation In the case α*<α<1, this extends a result of Brown, Wang and Wilson (1993).

We study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

In this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to an m-primary ideal I is effectively bounded by the degree of nilpotency of I. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

This paper is an excerpt from a Rayleigh essay submitted at the University of Cambridge in January 1970. We reproduce it now as it gives a general construction of a family of Moufang loops to which all bar one of the finite subloops of the Cayley algebra belong. These subloops were classified up to isomorphism in the original essay, but are classified up to equivalence under the action of the group of symmetries of in Boddington and Rumynin [1]. Explicitly, given a group G with an element a such that a2=1 in its centre, we construct a Moufang group in which G has index 2. will be non-associative unless G is abelian.

Let Φ be a root system and let Γ ⊆ Φ. In this short paper we prove that Γ contains a -basis of the lattice that it generates.

We consider the abelianizations of some normal subgroups of the automorphism group of a finitely generated free group. Let Fn be a free group of rank n. For d ≥ 2, we consider a group consisting the automorphisms of Fn which act trivially on the first homology group of Fn with /d-coefficients. We call it the congruence IA-automorphism group of level d and denote it by IAn,d. Let IOn,d be the quotient group of the congruence IA-automorphism group of level d by the inner automorphism group of a free group. We determine the abelianization of IAn,d and IOn,d for n ≥ 2 and d ≥ 2. Furthermore, for n=2 and odd prime p, we compute the integral homology groups of IA2,p for any dimension.

We establish Whitehead and Ganea characterizations for proper LS-category. We use the embedding of the proper category into the exterior category, and construct in the latter a suitable closed model structure of Strøm type. Then, from the axiomatic LS-category arising from the exterior homotopy category we can recover the corresponding proper LS invariants. Some applications are given.

We establish the existence and uniqueness of a factorization for geometric morphisms that generalizes the pure, complete spread factorization for geometric morphisms with a locally connected domain. A complete spread with locally connected domain over a topos is a geometric counterpart of a Lawvere distribution on the topos, and the factorization itself is of the comprehensive type. The new factorization removes the topologically restrictive local connectedness requirement by working with quasicomponents in topos theory. In the special case when the codomain topos of the geometric morphism coincides with the base topos, the factorization gives the locale of quasicomponents of the domain topos, or its ‘0-dimensional’ reflection.

Goussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.

In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

Using algebraic and topological K-theory together with complex C*-algebras, we prove that every abelian group may be realized as the centre of a strongly torsion generated group whose integral homology is zero in dimension one and isomorphic to two arbitrarily prescribed abelian groups in dimensions two and three.

Let V be a k-dimensional -vector space and let W be an n-dimensional vector subspace of V. Denote by GL(n, ) • 1k-n the subgroup of GL(V) consisting of all isomorphisms ϕ:V → V with ϕ(W) = W and ϕ(v) ≡ v (mod W) for every v ∈ V. We show that GL(3, ) • 1k-3 is, in some sense, the smallest subgroup of GL(V)≅ GL(k, , whose invariants are hit by the Steenrod algebra acting on the polynomial algebra, . The result is some aspect of an algebraic version of the classical conjecture that the only spherical classes inQ0S0are the elements of Hopf invariant one and those of Kervaire invariant one.

Let K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

In a fundamental paper in 1984, Vaughan Jones developed his new polynomialinvariant of knots using a Markov trace on the Temperley–Liebalgebra. Subsequently, Lickorish used the associated bilinear pairing toprovided an alternative proof for the existence of the 3-manifold invariants ofWitten, Reshetinkin and Turaev. A key property of this form is thenon-degeneracy of this form except at the parameter values±2cos(π/(n+1)) [7]. Koand Smolinsky derived a recursive formula for the determinants of specificminors of Markov's form, establishing the needed non-degeneracy[6]. In this paper,we define a triangular change of basis in which the form is diagonal andexplicitly compute the diagonal entries of this matrix as products of quotientsof Chebyshev polynomials, corroborating the determinant computation of Ko andSmolinsky. The method of proof employs a recursive method for defining therequired orthogonal basis elements in the Temperley–Lieb algebra,similar in spirit to Jones' and Wenzl's recursive formulafor a family of projectors in the Temperley–Lieb algebra. We define apartial order on the non-crossing chord diagram basis and give an explicitformula for a recursive construction of an orthogonal basis, via a recursionover this partial order. Finally we relate this orthogonal basis to basesconstructed using the calculus of trivalent graphs developed by Kauffman andLins [5].