Let $G$ be a locally compact group. The question of whether ${\cal H}^1(L^1(G),M(G))$, the first Hochschild cohomology group of $L^1(G)$ with coefficients in $M(G)$, is zero was first studied by B. E. Johnson and initiated his development of the theory of amenable Banach algebras. He was able to show that ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ whenever $G$ is amenable, a $[SIN]$-group, or a matrix group satisfying certain conditions. No group such that ${\cal H}^1(L^1(G),M(G)) \neq \{ 0 \}$ is known. In this paper, we approach the problem of whether ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ from several angles. Using weakly almost periodic functions, we show that ${\cal H}^1(L^1(G),L^1(G))$ is always Hausdorff for unimodular $G$. We also show that for $[IN]$-groups, every derivation $D \colon L^1(G) \to L^1(G)$ is implemented, not necessarily by an element of $M(G)$, but at least by an element of $\mbox{VN}(G)$, the group von Neumann algebra of $G$. This applies, in particular, to the group $G := {\mathbb T}^2 \rtimes \mbox{SL}(2, {\mathbb Z})$, for which it is unknown whether ${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$. Finally, we analyse the structure of derivations on $L^1(G)$; an important r\^ole is played by the closed normal subgroup $N$ of $G$ generated by the elements of $G$ with relatively compact conjugacy classes. We can write an arbitrary derivation $D \colon L^1(G) \to L^1(G)$ as a sum $D = D_N + D_{N^\perp}$, where $D_N$ and $D_{N^\perp}$ can be tackled with different techniques. Under suitable conditions, all satisfied by ${\mathbb T}^2 \rtimes \mbox{SL}(2,{\mathbb Z})$, we can show that $D_N$ is implemented by an element of $\mbox{VN}(G)$ and that $D_{N^\perp}$ is implemented by a measure. 1991 Mathematics Subject Classification: 22D05, 22D25, 43A10, 43A20, 46H25, 46L10, 46M20, 47B47, 47B48.

We consider quadratic diophantine equations of the shape

$$ Q(x_1, \ldots, x_s) = 0 \qquad (1)$$

for a polynomial $Q(X_1, \ldots, X_s) \in \mathbf{Z}[X_1, \ldots, X_s]$ of degree $2$. Let $H$ be an upper bound for the absolute values of the coefficients of $Q$, and assume that the homogeneous quadratic part of $Q$ is non-singular. We prove, for all $s \ge 3$, the existence of a polynomial bound $\Lambda_s(H)$ with the following property: if equation (1) has a solution $\mathbf{x} \in \mathbf{Z}^s$ at all, then it has one satisfying

$$ |x_i| \le \Lambda_s(H) \quad (1 \le i \le s). $$

For $s = 3$ and $s = 4$ no polynomial bounds $\Lambda_s(H)$ were previously known, and for $s \ge 5$ we have been able to improve existing bounds quite significantly.

We consider covering properties of weak topologies of Banach spaces, especially of weak or point-wise topologies of function spaces $C(K)$, for compact spaces $K$. We answer questions posed by A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Our main results are the following. A Banach space of density at most $\omega_1$ is hereditarily metaLindel\" of in its weak topology. If the weight of a compact space $K$ is at most $\omega_1$, then the spaces $C_w(K)$ and $C_p(K)$ are hereditarily metaLindel\" of. Let $\bar T$ be the one-point compactification of a tree $T$. Then the space $C_p(\bar T)$ is hereditarily $\sigma$-metacompact. If $T$ is an infinitely branching full tree of uncountable height and of cardinality bigger than ${\bf c}$, then the weak topology of the unit sphere of $C(\bar T)$ is not $\sigma$-fragmented by any metric. The space $C_p(\beta\omega_1)$ is neither metaLindel\" of nor $\sigma$-relatively metacompact. The space $C_p(\beta\omega_2)$ is not $\sigma$-relatively metaLindel\" of. Under the set-theoretic axiom $\diamondsuit$, there exists a scattered compact space $K_1$ such that the space $C_p(K_1)$ is not $\sigma$-relatively metacompact, and under a related axiom \boxd, there exists a scattered compact space $K_2$ such that the space $C_p(K_2)$ is not $\sigma$-relatively metaLindel\" of.

1991 Mathematics Subject Classification: 54C35, 46B20, 54E20, 54D30.

In the context of Mackey functors we introduce a category which is analogous to the category of modules for a quasi-hereditary algebra which have a filtration by standard objects. Many of the constructions which work for quasi-hereditary algebras can be done in this new context. In particular, we construct an analogue of the `Ringel dual', which turns out here to be a standardly stratified algebra. The Mackey functors which play the role of the standard objects are constructed in the same way as functors which have been used previously in parametrizing the simple Mackey functors, but instead of using simple modules in their construction (as was done before) we use p-permutation modules. These Mackey functors are obtained as adjoints of the operations of forming the Brauer quotient and its dual. The filtrations which have these Mackey functors as their factors are closely related to the filtrations whose terms are the sum of induction maps from specified subgroups, or are the common kernel of restriction maps to these subgroups. These latter filtrations appear in Conlon's decomposition theorems for the Green ring, as well as in other places, where they arise quite naturally.

2000 Mathematics Subject Classification: primary 20C20; secondary 20J05, 19A22, 16G70, 16E60.

Let $X$ be a regular separated scheme of finite Krull dimension and let $U^{n}_{X} \subset A^{n}_{X}$ be the punctured affine $n$-space over $X$. We show that the total graded Witt ring of $U^{n}_{X}$ is a free graded module over the total graded Witt ring of $X$ with two generators $1$ and $\epsilon$. The second generator satisfies the equation $\epsilon^{2} = 1$ when $n = 1$ and $\epsilon^{2} = 0$ when $n \geq 2$.

In the mid-1960s Borevi$\setminus$v\{c\} and Faddeev initiated the study of the Galois module structure of groups of \$p\$th-power classes of cyclic extensions \$K/F\$ of \$p\$th-power degree. They determined the structure of these modules in the case when \$F\$ is a local field. In this paper we determine these Galois modules for all base fields \$F\$.

Let $G$ be a finite group. We classify $G$-equivariant flow equivalence of non-trivial irreducible shifts of finite type in terms of

(i) elementary equivalence of matrices over $ZG$ and

(ii) the conjugacy class in $ZG$ of the group of G-weights of cycles based at a fixed vertex.

In the case $G = Z/2$, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of $E(ZG)$ equivalence, which involves $K_1(ZG)$.

Using Grothendieck's semicontinuity theorem for half-exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact sequence relating the first Hochschild cohomology group of an algebra to that of the endomorphism algebra of a module and apply it to study the first Hochschild cohomology group of an Auslander algebra. Our main result shows that for a finite-dimensional and representation-finite algebra algebra $A$ over an algebraically closed field with Auslander algebra $\Lambda$ the following conditions are equivalent:

(1) $A$ admits no outer derivation;

(2) $\Lambda$ admits no outer derivations;

(3) $A$ is simply connected;

(4) $\Lambda$ is strongly simply connected.

Let $G$ be a semisimple algebraic group defined over an algebraically closed field $K$ of good characteristic $p>0.$ Let $u$ be a unipotent element of $G$ of order $p^{t}$, for some $t\in {\Bbb N}$. In this paper it is shown that $u$ lies in a closed subgroup of $G$ isomorphic to the {\it Witt group} $W_{t}(K),$ which is a $t$-dimensional connected abelian unipotent algebraic group. 2000 Mathematics Subject Classification: 20G15.

It is shown that the topologically irreducible representations of a normed algebra define a certain topological radical in the same way that the strictly irreducible representations define the Jacobson radical and that this radical can be strictly smaller than the Jacobson radical. An abstract theory of ‘topological radicals’ in topological algebras is developed and used to relate this radical to the Baer radical (prime radical). The relations with topologically transitive representations and standard representations in the sense of Meyer are also explored.

1991 Mathematics Subject Classification: 46H15, 46H25, 16Nxx.

In order to apply the ideas of Iwasawa theory to the symmetric square of a newform, we need to be able to define non-archimedean analogues of its complex $L$-series. The interpolated $p$-adic $L$-function is closely connected via a “Main Conjecture” with certain Selmer groups over the cyclotomic $\Bbb Z_p$-extension of $\Bbb Q$. In the $p$-ordinary case these functions are well understood.

In this article we extend the interpolation to an arbitrary set $S$ of good primes (not necessarily satisfying ordinarity conditions). The corresponding $S$-adic functions can be characterised in terms of certain admissibility criteria. We also allow interpolation at particular primes dividing the level of the newform.

One interesting application is to the symmetric square of a modular elliptic curve $E$ defined over $\Bbb Q$. Our constructions yield $p$-adic $L$-functions at all primes of stable or semi-stable reduction. If $p$ is ordinary or multiplicative the corresponding analytic function is bounded; if $p$ is supersingular our function behaves like $\log^2(1+T)$.

1991 Mathematics Subject Classification: 11F67, 11F66, 11F33, 11F30

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

The Jucys–Murphy elements for wreath products $\Gamma_n = \Gamma \wr S_n $ associated to any finite group $\Gamma$ are introduced and they play an important role in our study of the connections between class algebras of $\Gamma_n$ for all $n$ and vertex algebras. We construct an action of (a variant of) the $\mathcal W_{1 + \infty}$-algebra acting irreducibly on the direct sum ${\mathcal R}_\Gamma$ of the class algebras of $\Gamma_n$ for all $n$ in a group-theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on ${\mathcal R}_\Gamma$. As applications, we obtain two distinct sets of algebra generators for the class algebra of $\Gamma_n$ and establish various stability results concerning products of normalized conjugacy classes of $\Gamma_n$ and the power sums of Jucys--Murphy elements, etc. We introduce a stable algebra which encodes the class algebra structures of $\Gamma_n$ for all $n$, whose structure constants are shown to be non-negative integers. In the symmetric group case (that is, when $\Gamma$ is trivial), we recover and strengthen in a uniform approach various results of Lascoux and Thibon, Kerov and Olshanski, and Farahat and Higman, etc.

Let $k\ge 3$ be an integer. For $0<s<1$, let $D_s \subset {\Bbb R}^2$ be the set that is constructed iteratively as follows. Take a regular open $k$-gon with sides of unit length, attach regular open $k$-gons with sides of length $s$ to the middles of the edges, and so on. At each stage of the iteration the $k$-gons that are added are a factor $s$ smaller than the previous generation and are attached to the outer edges of the family grown so far. The set $D_s$ is defined to be the interior of the closure of the union of all the $k$-gons. It is easy to see that there must exist some $s_k>0$ such that no $k$-gons overlap if and only if $0<s\le s_k$. We derive an explicit formula for $s_k$.

The set $D_s$ is open, bounded, connected and has a fractal polygonal boundary. Let $E_{D_s}(t)$ denote the heat content of $D_s$ at time $t$ when $D_s$ initially has temperature $0$ and $\partial D_s$ is kept at temperature $1$. We derive the complete short-time expansion of $E_{D_s}(t)$ up to terms that are exponentially small in $1/t$. It turns out that there are three regimes, corresponding to $0<s<1/(k-1)$, $s=1/(k-1)$, and $1/(k-1)<s\le s_k$. For $s \neq 1/(k-1)$ the expansion has the form

$$ E_{D_s}(t)=p_s(\log t)t^{1-d_s/2}+A_st^{1/2}+Bt+O(e^{-r_s/t}), $$

where $p_s$ is a $\log (1/s^2)$-periodic function, $d_s=\log (k-1)/\log (1/s)$ is a similarity dimension, $A_s$ and $B$ are constants related to the edges and vertices, respectively, of $D_s$, and $r_s$ is an error exponent. For $s=1/(k-1)$, the $t^{1/2}$-term carries an additional $\log t$.

We investigate asphericity of the relative group presentation $$\langle G,t\mid atbtctdtet=1\rangle$$ and prove it aspherical provided the subgroup of $G$ generated by $$\{ ab^{-1}, bc^{-1}, cd^{-1}, de^{-1}\}$$ is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation $$\langle G,s,t \mid \alpha s\beta s\gamma t= 1=\delta t\varepsilon t\zeta s^{-1}\rangle. 2000 Mathematics Subject Classification: 20F05, 57M05.

First, we prove that a non-abelian normal CM-field of degree 16 has odd relative class number if and only if it is dihedral, or is a compositum of two normal octic CM-fields with the same maximal real subfield, or has Galois group $G_9 =\langle a,b,z: a^2=b^2=z^4=1,b^{-1}ab=az^2,az=za,bz=zb\rangle$. Then, we solve several relative class number one problems. (1) We solve the relative class number one problem for the dihedral CM-fields of 2-power degrees. First, we remind the reader of the characterization of the dihedral CM-fields of 2-power degrees with odd relative class numbers. Second, we give lower bounds on relative class numbers of dihedral CM-fields of 2-power degrees with odd relative class numbers. We thus obtain an upper bound on the discriminants of the dihedral CM-fields of 2-power degrees with relative class number equal to 1. Third, we compute the relative class numbers of all the dihedral CM-fields of 2-power degrees with odd relative class numbers and discriminants less than or equal to this latter bound. We end up with a list of twenty-four dihedral CM-fields of 2-power degrees with relativeclass numbers equal to 1, and show that exactly twenty-one of them have class number 1. (2) We determine all the non-abelian normal CM-fields of degree 16 with Galois group $G_9$ which have relative class number 1 (there is only one such number field), and then those which have class number 1 (there is only one such number field). (3) We determine some of the non-abelian normal CM-fields $N =N_1N_2$ of degree 16 which are composita of two normal octic CM-fields $N_1$ and $N_2$ with the same maximal real subfield which have relative class number 1, and then those which have class number 1. Indeed, we focus on the case where one of the $N_i$ is a quaternion octic CM-field and prove that there is only one such compositum with relative class number 1 and that this compositum has class number 1.

1991 Mathematics Subject Classification: primary 11R29; secondary 11R21, 11R42, 11M20, 11Y40.

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition $\lambda$, we find a presentation of the Specht module $S^{\lambda}$ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of $S^{\lambda}$ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape $\lambda$.

This series of papers presents and rigorously analyzes a probabilistic algorithm for finding small prime factors of an integer. The algorithm uses the Jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. This second paper in the series is concerned with the order of the group of rational points on the Jacobian of a curve of genus 2 defined over a finite field. We prove a result on the distribution of these orders.

2000 Mathematical Subject Classification: 11Y05, 11G10, 11M20, 11N25.

Let $T$ be a contraction on a complex Hilbert space $H$. A result of Brown, Chevreau and Pearcy from 1979 shows that $T$ has a non-trivial invariant subspace if the spectrum of $T$ is dominating in the open unit disc. It is the purpose of the present paper to prove the multidimensional analogue of this result for spherical contractions $T \in L(H)^n$ that possess a spherical dilation and for which the Harte spectrum is dominating in the open unit ball $B$ in $\mathbb{C}^n$. If even the essential Harte spectrum of $T$ is dominating in $B$, then $T$ is shown to be reflexive and to possess an extremely rich invariant subspace lattice. The proof is based on the existence of an $H^{\infty}$-functional calculus for completely non-unitary spherical contractions and on a multidimensional analogue of the classical result of Sz.Nagy and Foias, stating that each spherical contraction which is neither of type $C_{\cdot 0}$ nor of type $C_{0 \cdot}$ and which does not consist of multiples of the identity operator on $H$, possesses non-trivial joint hyperinvariant subspaces.

1991 Mathematics Subject Classification: 47A13, 47A15, 47A60