We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.

In this paper we extend a well-known theorem of M.~Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

If an operator $T$ satisfies a modular inequality on a rearrangement invariant space $L^\rho (\Omega,\mu)$, and if $p$ is strictly between the indices of the space, then the Lebesgue inequality $\int |Tf|^p \leq C \int |f|^p$ holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form $\int \Phi (|Tf|) \leq \int \Phi (C |f|)$, and here, one can extrapolate to the (finite) indices $i(\Phi)$ and $I(\Phi)$ as well.

Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains $\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there is a unique normalised generator of the field of $G$-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.

A local version of $\VMO$ is defined, and the local Hardy space $\hone$ is shown to be its dual. An application to weak-$*$ convergence in $\hone$ is proved.

Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov's question negatively.

We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.

On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.

Let $k$ be a cyclic extension of odd prime degree $p$ of the field of rational numbers. If $t$ denotes the number of primes that ramify in $k$, it is known that the Hilbert $p$-class field tower of $k$ is infinite if $t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive proportion of such fields $k$ have infinite Hilbert $p$-class field towers.

Using a classical generalization of Jacobi's derivative formula, we give an explicit expression for Gunning's prime form in genus 2 in terms of theta functions and their derivatives.

We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real M\"obius transformations. Included are the maps that are exactly $n$-to-$1$, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

D.~H.~Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P.~Erd\H os, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.

Let $b$ be an integer with $b>1$. In this note, we prove that the number of non-zero digits in the base $b$ representation of $n!$ grows at least as fast as a constant, depending on $b$, times $\log n$.

Sufficient conditions are given in order to prove the lowest unknown case of the grade conjecture. The proof combines vanishing results of local cohomology and the $S_{2}$ condition.

We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the `physical') space and its internal space. We prove, assuming only that the window defining the model set is measurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

We give a topological interpretation of the core group invariant of a surface embedded in $S^4$ \cite{F-R}, \cite{Ro}. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of $S^4$ with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres.

Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be a root of $X^5 + aX + b$. An explicit formula is given for the discriminant of $Q(\theta)$.

Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

Si $M$ est une vari\'et\'e de dimension $n$, compacte non simplement connexe, on caract\'erise les m\'etriques riemanniennes sur $M$ dont la fonction croissance a exactement deux singularit\'es.

We explicitly describe, in terms of indecomposable $\mathbb{Z}_2 [G]$-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completes work begun in [Elder: Canad. J.~Math. (5) {\bf 50}(1998), 1007--1047].

It is known that the $K$-theory of a large class of groups can be computed from the $K$-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the $K$-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of ${\nk}_0$-groups that appear in the calculation of the $K_0$-groups of virtually infinite cyclic groups.

We prove that the Mahler measure of an algebraic number cannot be too close to an integer, unless we have equality. The examples of certain Pisot numbers show that the respective inequality is sharp up to a constant. All cases when the measure is equal to the integer are described in terms of the minimal polynomials.

We compute the rational Chow groups of supersingular abelian varieties and some other related varieties, such as supersingular Fermat varieties and supersingular $K3$ surfaces. These computations are concordant with the conjectural relationship, for a smooth projective variety, between the structure of Chow groups and the coniveau filtration on the cohomology.

The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group $\Gr^2 (A_{\bar{k}})$ of a supersingular abelian variety $A_{\bar{k}}$ over the algebraic closure of a finite field of characteristic $p$ is at most a $p$-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of C.~Schoen it is also shown that if the Tate conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field $k$ of characteristic $p>2$, then the Griffiths group of any ordinary abelian threefold $A_{\bar{k}}$ over the algebraic closure of $k$ is non-trivial; in fact, for all but a finite number of primes $\ell\ne p$ it is the case that $\Gr^2 (A_{\bar{k}}) \otimes \Z_\ell \neq 0$.

An irreducible supercuspidal representation $\pi$ of $G= \GL(n,F)$, where $F$ is a nonarchimedean local field of characteristic zero, is said to be ``distinguished'' by a subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if $\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue of this notion for and irreducible, automorphic, cuspidal representation associated to $\GL(n)$. Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

Erratum to {\it The Lehmer Polynomial and Pretzel Links}, Canad. J. Math. {\bf 44}(2001), 440--451.

The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

We combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles' deep machinery.

We use the theory of Jacobi-like forms to construct modular forms for a congruence subgroup of $\SL(2,\mathbb{R})$ which can be expressed as linear combinations of products of certain theta functions.

H.~O.~Kim has shown that contrary to the case of $H^p$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, {\it i.e.} they have the same dual spaces and the same Fr\'echet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into $H^p$, $0 < p \leq \infty$.

In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals.

A new construction of the ordinary residue of differential forms is given. This construction is intrinsic, \ie, it is defined without local coordinates, and it is geometric: it is constructed out of the geometric structure of the local and global cohomology groups of the differentials. The Residue Theorem and the local calculation then follow from geometric reasons.

We study a class of subgroups of $\PSL_2 (\mathbb{Z})$ which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.

A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra $\calT = \calT (L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT (\calL)$ of $\calT$ which has the same essential commutant as that of $\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$.

The Cuntz-Krieger algebra $\mathcal{O}_B$ is defined for an arbitrary, possibly infinite and infinite valued, matrix $B$. A graph $C^{\ast}$-algebra $G^{\ast} (E)$ is introduced for an arbitrary directed graph $E$, and is shown to coincide with a previously defined graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the Cuntz-Krieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix of~$E$.

For a modular elliptic curve $E/\mathbb{Q}$, we show a number of links between the primes $\ell$ for which the mod $\ell$ representation of $E/\mathbb{Q}$ has projective dihedral image and congruence primes for the newform associated to $E/\mathbb{Q}$.

Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some algebraically closed field $k$ of characteristic $0$. For an integer $t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the dimension of the space of global sections of $O(t)$, and let $k$ be an integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$ be general points on $\mathbf{P}^n$ and let $\pi \colon X \to \mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points. Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 - \cdots -E_k)$ is a very ample invertible sheaf on $X$.

On explicite une classe de champ de vecteurs polynomiaux non analytiquement lin\'earisables \`a l'aide de la correction introduite par \'Ecalle-Vallet. Notamment, on \'etend des r\'esultats de Schuman sur la trivialit\'e des hamiltoniens homog\`enes isochrones. We characterize a class of polynomial vector fields which are not analytically linearizable using the correction introduced by \'Ecalle-Vallet. Then, we extend Schuman's result about non existence of isochronous homogenous Hamiltonian systems.

In this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.

A partial differential equation, the local M\"obius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local M\"obius equation and an additional partial differential equation.

As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree~5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type~$E_8$.

Exact analytical expressions for the inverse Laplace transforms of the functions $\frac{I_n(s)}{s I_n^\prime(s)}$ are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite denumerable set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author's knowledge, is new, as the inverse transforms of $\frac{I_n(s)}{s I_n^\prime(s)}$ have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order $n$. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the sheaf of germs of holomorphic sections of the tangent bundle $TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all $q \geq 0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on $M_n$ consists of a point. Thus the complex structure on $M_n$ is locally rigid.

We give an upper bound on the essential dimension of the group $\mathbb{Z}/q\rtimes(\mathbb{Z}/q)^*$ over the rational numbers, when $q$ is a prime power.

The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in \mathbb{N}$ is not a perfect square. This continues work in \cite{me}--\cite{alfnme}.

In this paper, we give an integral formula for the eigenfunctions of the ring of differential operators related to the root system $B_2$.

An error in {\it A characterization of left perfect rings}, Canad. Math. Bull. (3) {\bf 38}(1995), 382--384, is indicated and the consequences identified.

In this paper we study simple associative algebras with finite $\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$ that has been constructed in Morita theory from a bilinear form $g\colon U\times V\to A$ over a simple algebra $A$. We show that finite $\mathbb{Z}$-gradings on $F_g$ are in one to one correspondence with certain decompositions of the pair $(U,V)$. We also show that any simple algebra $R$ with finite $\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some bilinear from $g\colon U\times V \to A$, where the grading on $F_g$ is determined by a decomposition of $(U,V)$ and the coordinate algebra $A$ is chosen as a simple ideal of the zero component $R_0$ of $R$. In order to prove these results we first prove similar results for simple algebras with Peirce gradings.

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice $\varGamma$ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diffraction measure is periodic, with the dual lattice $\varGamma^*$ as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.

Let $\Phi$ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$-grading on the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p (\Phi)$ being a graded division algebra, while the grading of $C\cong M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical ``induction'' procedure.

We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type $C(n)$, $D(m,n)$, $D(2,1;\alpha)$ $(\alpha \in \mathbb{F} \setminus \{0,-1\})$, $F(4)$, and $G(3)$.

We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.

No abstract.

We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that are normalized by the sl$_2$-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The connection goes both ways---we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.

We begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using `postcards') a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity.

We construct a class of fermions (or bosons) by using a Clifford (or Weyl) algebra to get two families of irreducible representations for the extended affine Lie algebra $\widetilde{\mathfrak{gl}_N (\mathbb{C}_q)}$ of level $(1,0)$ (or $(-1,0)$).

This paper characterizes when a Delone set $X$ in $\mathbb{R}^n$ is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set $X$, let $N_X (T)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let $M_X(T)$ be the minimum radius such that every closed ball of radius $M_X(T)$ contains the center of a patch of every one of these kinds. We show that for each of these functions there is a ``gap in the spectrum'' of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to $X$ being an ideal crystal. Explicitly, for $N_X(T)$, if $R$ is the covering radius of $X$ then either $N_X(T)$ is bounded or $N_X (T) \ge T/2R$ for all $T>0$. The constant $1/2R$ in this bound is best possible in all dimensions. For $M_X(T)$, either $M_X(T)$ is bounded or $M_X(T)\ge T/3$ for all $T>0$. Examples show that the constant $1/3$ in this bound cannot be replaced by any number exceeding $1/2$. We also show that every aperiodic Delone set $X$ has $M_X(T)\ge c(n) T$ for all $T>0$, for a certain constant $c(n)$ which depends on the dimension $n$ of $X$ and is $>1/3$ when $n>1$.

In this paper we study specializations and one-sided bimodules of simple Jordan superalgebras.

We study the representations of extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$-th primitive root of unity ($\mathbb{C}_q$ is the quantum torus in two variables). We first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a suitable number of copies is a quotient of $s\ell_{\ell+1} (\mathbb{C}_q)$. Thus any finite dimensional irreducible module for $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of $s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$ comes from above. We then construct modules for the extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1 \oplus \mathbb{C} d_2$ which is integrable and has finite dimensional weight spaces.

We embed the moduli space $Q$ of 5 points on the projective line $S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the 6-dimensional irreducible module of the symmetric group $S_5$. This module splits with respect to the icosahedral group $A_5$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of $A_5$ on $Q$ to those on the regular icosahedron.

We consider two dynamical systems associated with a substitution of Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and the $\mathbb{R}$-action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the ``balanced pairs algorithm'' and the ``overlap algorithm'') and study the relation between them. In particular, we show that pure discrete spectrum for the $\mathbb{R}$-action implies pure discrete spectrum for the $\mathbb{Z}$-action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every $\mathbb{R}$-action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).

Quantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type $\rmC$.

No abstract.