Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler–Poincaré characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [Ascr ], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [Ascr ] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts.

Our formula for the equivariant Euler characteristic over [Ascr ] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.

As a special case of a well-known conjecture of Artin, it is expected that a system of R additive forms of degree k, say

formula here

with integer coefficients aij, has a non-trivial solution in ℚp for all primes p whenever

formula here

Here we adopt the convention that a solution of (1) is non-trivial if not all the xi are 0. To date, this has been verified only when R=1, by Davenport and Lewis [4], and for odd k when R=2, by Davenport and Lewis [7]. For larger values of R, and in particular when k is even, more severe conditions on N are required to assure the existence of p-adic solutions of (1) for all primes p. In another important contribution, Davenport and Lewis [6] showed that the conditions

formula here

are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N[Gt ]R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints

formula here

to be sufficient for p-adic solubility of (1).

A noticeable feature of these results is that for even k, one always encounters a factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can reach the expected order of magnitude k2.

We characterize composition operators on spaces of real analytic functions which at the same time have closed image and are open onto their images. Under some mild assumptions, we also characterize composition operators with closed range and composition operators open onto their images.

The Hall–Paige conjecture deals with conditions under which a finite group $G$ will possess a complete mapping, or equivalently a Latin square based on the Cayley table of $G$ will possess a transversal. Two necessary conditions are known to be: (i) that the Sylow 2-subgroups of $G$ are trivial or non-cyclic, and (ii) that there is some ordering of the elements of $G$ which yields a trivial product. These two conditions are known to be equivalent, but the first direct, elementary proof that (i) implies (ii) is given here.

It is also shown that the Hall–Paige conjecture implies the existence of a duplex in every group table, thereby proving a special case of Rodney's conjecture that every Latin square contains a duplex. A duplex is a ‘double transversal’, that is, a set of $2n$ entries in a Latin square of order $n$ such that each row, column and symbol is represented exactly twice.

This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as $l^1(q)$ , and this definition is extended to all height $m$ polynomials as \[ l^m(q):= \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n,\, \epsilon_i \in {\bb Z},\, \vert\epsilon_i\vert \leqslant m,\, y \ne 0). \] A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of $l^m(q)$ when $q$ is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows.

THEOREM. Let $q$ be a quadratic Pisot number that satisfies a polynomial of the form $p(x) = x^2 - ax \pm 1$ , with conjugate $r$ . Let $q$ have convergents $\{C_k/D_k\}$ and let $k$ be the maximal integer such that \[ \vert D_k r - C_k\vert \leqslant m \frac{1}{1-\vert r\vert}; \] then \[ l^m (q) = \vert D_k q - C_k \vert. \]

A value related to $l(q)$ is $a(q)$ , the minimal non-zero value when all ${\pm}1$ polynomials are evaluated at $q$ . Formally, this is \[ a(q) := \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^2 + \cdots + \epsilon_n q^n,\, \epsilon_i = {\pm} 1,\, y \ne 0). \] An open question concerning how often $a(q) = l(q)$ is also answered in this paper.

Let $A_n=K\langle x_1,\ldots,x_n \rangle$ be a free associative algebra over a field $K$. In this paper, examples are given of elements $u \in A_n$, $n \ge 3$, such that the factor algebra of $A_n$ over the ideal generated by $u$ is isomorphic to $A_{n-1}$, and yet $u$ is not a primitive element of $A_n$ (that is, it cannot be taken to $x_1$ by an automorphism of $A_n$). If the characteristic of the ground field $K$ is $0$, this yields a negative answer to a question of G. Bergman. On the other hand, by a result of Drensky and Yu, there is no such example for $n=2$. It should be noted that a similar question for polynomial algebras, known as the embedding conjecture of Abhyankar and Sathaye, is a major open problem in affine algebraic geometry.

A mean-value theorem, an inverse mapping theorem and an implicit mapping theorem are established here in a class of complex locally convex spaces, including the spaces of test functions in distribution theory. Our main tool is the integral formula and the invariance of the domain, derived from topological degrees, rather than from fixed points of contractions in Banach spaces.

Let Γ be a Fuchsian group. We denote by ax(Γ) the set of axes of hyperbolic elements of Γ. Define Fuchsian groups Γ1 and Γ2 to be isoaxial if ax(Γ1)=ax(Γ2). The main result in this note is to show (see Section 2 for definitions) the following.

Theorem 1.1. Let Γ1and Γ2be isoaxial arithmetic Fuchsian groups. Then Γ1and Γ2are commensurable.

This result was motivated by the results in [6], where it is shown that if Γ1 and Γ2 are finitely generated non-elementary Fuchsian groups having the same non-empty set of simple axes, then Γ1 and Γ2 are commensurable. The general question of isoaxial was left open. Theorem 1.1 is therefore a partial answer. Arithmetic Fuchsian groups are a very special subclass of Fuchsian groups (for example, there are at most finitely many conjugacy classes of such groups of a fixed signature), and the general case at present seems much harder to resolve.

The technical result of this paper which implies Theorem 1.1 is Theorem 2.4 (below), proved in Section 4. Denoting the commensurability subgroup by Comm(Γ) and the subgroup of PGL(2, R) which preserves the set ax(Γ) by Σ(Γ) (careful definitions are given below), we have the following theorem which describes the commensurability subgroup of an arithmetic Fuchsian group geometrically.

Theorem 2.4. Let Γ be an arithmetic Fuchsian group. Then Comm(Γ)=Σ(Γ).

The problem of ‘isoaxial implies commensurable’ is like a dual problem to ‘isospectral implies commensurable’. It was shown in [8] that this latter statement holds for arithmetic Fuchsian groups. As in the case considered here, the general statement about isospectrality implying commensurability is still open.

Wirtinger and Northcott introduced an inequality bounding the norm of a 2$\Omega$ periodic function by the norm of its derivatives. It is shown here that this type of result occurs for other operators, defined on various domains and spaces.

For any x ∈ (0, 1], let the series [sum ]∞n=1 1/dn(x) be the Sylvester expansion of x. Galambos has shown that the Lebesgue measure of the set

[formula here]

is 1 when α = e, the base of the natural logarithm. This paper provides a proof that for any α [ges ] 1, A(α) has Hausdorff dimension 1 when α ≠ e.

The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. An analogous problem concerning spreading models is also addressed.

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S, at least two distinct words of length n on these letters are equal in S. In particular, S is collapsing whenever it satisfies a law. Let [Uscr ](A) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: [Uscr ](A) is collapsing; [Uscr ](A) satisfies some semigroup law; [Uscr ](A) satisfies the Engel condition; [Uscr ](A) is nilpotent; A is nilpotent when considered as a Lie algebra.

a conjecture is proposed, bounding the number of cycles with label $w^n$ in a labeled directed graph. some partial results towards this conjecture are established. as a consequence, it is proved that $\langle a_1, a_2, \ldots\,{\mid}\,w^n\rangle$ is coherent for $n\,{\geq}\,4$. furthermore, it is coherent for $n\,{\geq}\,2$, provided that the strengthened hanna neumann conjecture holds.

It is shown that a Carter subgroup of a non-soluble unitary group can only be the normalizer of a Sylow 2-subgroup. This result, combined with previous results, implies that no finite simple group can be a minimal counterexample to the conjugacy conjecture of Carter subgroups in a finite group.The third author was partially supported by the University of Padova, the RFBR, grants 02-01-00495 and 02-01-06226, Ministry of Education RF, E00-1.0-77, ‘Universities of Russia’ UR.04.01.031.

Let A be an order integral over a valuation ring V in a central simple F-algebra, where F is the fraction field of V. We show that (a) if (Vh, Fh) is the Henselization of (V, F), then A is a semihereditary maximal order if and only if A[otimes ]VVh is a semihereditary maximal order, generalizing the result by Haile, Morandi and Wadsworth, and (b) if J(V) is a principal ideal of V, then a semihereditary V-order is an intersection of finitely many conjugate semihereditary maximal orders; if not, then there is only one maximal order containing the V-order.

In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations $Sq^k$ on the cohomology ring of any cocommutative Hopf algebra over $Z/2$. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that $Sq^0$ is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra $\cal {B}$ that is generated by elements $Sq^k$ and subject to Adem relations; it shows that the Cartan formula gives a well-defined coproduct on $\cal {B}$. Also, it is shown that $\cal {B}$ with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper.

In this paper we use the Hausdorff metric to prove that two compact metric spaces are homeomorphic if and only if their canonical complements are uniformly homeomorphic. We take one of the two steps needed to prove that the difference between the homotopical and topological classifications of compact connected ANRs depends only on the difference between continuity and uniform continuity of homeomorphisms in their canonical complements, which are totally bounded metric spaces. The more important step was provided by the Chapman complement and the Curtis–Schori–West theorems. We also improve the multivalued description of shape theory given by J. M. R. Sanjurjo but only in the class of locally connected compacta.

It is proved that Epstein's zeta-function $\zeta_Q(s)$, related to a positive definite integral binary quadratic form, has a zero $ 1/2 +i\gamma $ with $T \leq \gamma \leq T+T^{5/11+\varepsilon }$ for sufficiently large positive numbers $T$. This improves a classical result of H. S. A. Potter and E. C. Titchmarsh (Proc. London Math. Soc. (2) 39 (1935) 372–384).

The concept of Morita equivalence is generalized to the context of locally C$^{*}$-algebras. This generalizes a well-known theorem of Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.

The question of which Riemannian manifolds admit simple closed geodesics is still a mystery. It is not known whether all closed Riemannian manifolds contain simple closed geodesics. For closed manifolds with nontrivial fundamental group, a simple closed geodesic can always be found by taking the shortest homotopically nontrivial closed geodesic. When the manifold is closed but simply connected, the question is open for dimensions three and above. In dimension two, it is known by the theorem of Lusternik and Schnirelmann [6] (see also [3] and [4]) that the 2-sphere equipped with any smooth Riemannian metric contains at least three distinct simple closed geodesics.

Non-compact manifolds do not necessarily contain closed geodesics, Euclidean space being an obvious example. Even if the manifold is not simply connected, it may not contain any simple closed geodesics, as with the hyperbolic thrice-punctured sphere. However, among the orientable, finite area, complete hyperbolic 2-manifolds, the thrice-punctured sphere is the only example that contains no simple closed geodesic. In this paper, we shall determine which orientable hyperbolic 3-manifolds do and do not contain simple closed geodesics. We shall prove that the Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite, or even that it has finitely generated fundamental group. The simple closed geodesic which we produce arises from an interesting class of elements of the fundamental group. It is the shortest closed geodesic corresponding to a screw motion induced by the action of the fundamental group on hyperbolic 3-space.

In proving our result, we use geometric methods to obtain results about isometries of hyperbolic 3-space. These results apply to elliptic, as well as to parabolic and loxodromic isometries, and we state them in full generality.

A related question is whether a hyperbolic 3-manifold always contains a non-simple closed geodesic. In [2], Alan Reid and Ted Chinburg utilized arithmetic hyperbolic 3-manifold theory to construct examples of closed hyperbolic 3-manifolds in which every closed geodesic is simple.

We shall be working with orientable 3-manifolds, so all of the isometries discussed will be orientation preserving.