Let $A$ be a commutative, cocommutative Hopf algebra, finitely generated and projective over its base ring $R$. Waterhouse asked whether the image of the class-invariant map, taking each $A$-Galois algebra to the class in ${\rm Pic}(A)$ of its $R$-linear dual, is the group of primitive classes in ${\rm Pic}(A)$. We discuss functorial aspects of this problem, and relate it to Fr\"ohlich's Hom-description of ${\rm Pic}(A)$ in the case that $R$ is a Dedekind domain with field of fractions $K$, and $A$ is an $R$-Hopf order in a separable $K$-Hopf algebra. We then apply this machinery to a certain class of Hopf orders $\mathfrak A$ in the Hopf algebra ${\rm Map}(G,K)$. More precisely, we give a positive answer to Waterhouse's question for $\mathfrak A$ when the dual $\mathfrak B$ of $\mathfrak A$ is one of the Hopf orders in $KG$ constructed by Larson, and a compatibility condition holds between the filtrations of $G$ determined by the various completions of $\mathfrak B$.

1991 Mathematics Subject Classification: 11R33, 16W30

To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function $$ F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), $$ where $\alpha$ is real and $T\ge 2$, $\gamma$ and $\gamma '$ denote the imaginary parts of zeros of the Riemann zeta-function, and $w(u) = 4/(4 + u^2)$. Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for $F(\alpha)$ when $|\alpha|\le 1$, and made the conjecture that $F(\alpha) = 1 + o(1)$ as $T\to \infty$ for any bounded $\alpha$ with $|\alpha |\ge 1$. In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions, then for any $\epsilon >0$ we have $$ F(\alpha) \ge {3\over 2} - |\alpha| - \epsilon ,$$ uniformly for $1\le |\alpha| \le \frac32 -2\epsilon $ and all $T \ge T_0(\epsilon)$.

1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.

The notion of \emph{hyperdecidability} has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly verified property of the first factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free profinite objects). As an application of the main results, the decidability of the Krohn--Rhodes (group) complexity is shown to follow from non-algorithmic abstract properties likely to be satisfied by the pseudovariety of all finite aperiodic semigroups, thereby suggesting a new approach to answer (affirmatively) the question as to whether complexity is decidable.

1991 Mathematics Subject Classification: primary 20M05, 20M07, 20M35; secondary 08B20.

In this paper we study the rational representation theory of the general linear group $G = \mbox{GL}_n(F)$ over an algebraically closed field $F$ of characteristic $p$. Given $\alpha \in {\Bbb Z} / p{\Bbb Z}$, we define functors $\mbox{Tr}^\alpha$ and $\mbox{Tr}_\alpha$, which, roughly speaking, are given by tensoring with the natural $G$-module $V$ and its dual $V^*$ respectively, and then projecting onto certain blocks determined by the residue $\alpha$. In fact, these functors can be viewed as special cases of Jantzen's translation functors. We prove a number of fundamental properties about these functors and also certain closely related functors that arise in the modular representation theory of the symmetric group.

1991 Mathematics Subject Classification: 20G05, 20C05.

In this paper we determine the possible Hilbert functions of a Cohen--Macaulay local ring of dimension $d$ and multiplicity $e$, in the case where the embedding dimension $v$ satisfies $v=e+d-3$ and the Cohen--Macaulay type is less than or equal to $e-3$.

1991 Mathematics Subject Classification: primary 13D40; secondary 13P99.

We find Zariski pairs of sextics with simple singularities having maximal total Milnornumber. We also relate them to the existence of distinct Mordell--Weil groups of extremal elliptic $K3$ surfaces with a fixed set of semistable singular fibres.

1991 Mathematics Subject Classification: 14F45, 14F27, 14F28.

Denote by $\phi_{\beta,p}$ the normal state on${\cal B}( \ell ^2 ({\Bbb N}))$ with list of eigenvalues $\lambda_{\beta ,p,k} =p^{-\beta k} (1-p^{-\beta} )$, where $k \in {\Bbb N}$, $\beta \in (0,1]$and $p$ is a prime number. For each $\beta $ and each subset ${\cal S}$ of the set ${\cal P}$ of all prime numbers, denote by $M_{\beta ,{\cal S}}$ the ITPFI factor (infinite tensor product of factors of type I)$$ \bigotimes\limits_{ p\in {\cal S}} \big( {\cal B}(\ell ^2 ({\Bbb N} )) ,\phi_{\beta,p} \big).$$ We prove that for any $\beta \in (0,1]$ and $\lambda \in [ 0,1]$, there exists ${\cal S}={\cal S}_{\beta,\lambda}$, asubset of ${\cal P}$, such that $M_{\beta, {\cal S}}$ isof type III$_\lambda$. Moreover, $M_{\beta ,{\cal P}}$ is the Araki--Woods factor $R_\infty$ for any $\beta \in (0,1]$. For any $\beta \in (0.5,1]$ and ${\cal S} \subset {\cal P}$, we show that $M_{\beta ,{\cal S}}$ is an ITPFI$_2$ factor. We study the class of Connes' T-groups of type III factors of the form $M_{\beta,{\cal S}}$, where ${\cal S} \subset {\cal P}$, which we call $\beta$-representable subgroups of ${\Bbb R}$ and show that it is rich, proving for each $\beta \in (0,1]$ that if $H$ is a countable subgroup of ${\Bbb R}$and $\Sigma$ a countable subset of ${\Bbb R} \setminus H$, there exists a $\beta$-representable group $\Gamma$ which contains $H$ and does not intersect $\Sigma$.Moreover, we prove that any $\beta_0$-representable group is $\beta$-representable if $1\geq \beta_0 \geq \max (\beta ,0.07)$ and that the class of $1$-representable groups is closed under the natural action of ${\Bbb R}^*_+$on the subgroups of ${\Bbb R}$. Our results show in particular that all hyperfinite type III$_\lambda$ factors, with $\lambda \in (0,1]$, and a large class of type III$_0$ ITPFI factors are isomorphic to group von Neumann algebras associated with some restricted adelic productsof affine motion groups of $p$-adic lines.

1991 Mathematics Subject Classification: 46L35, 11N05.

We prove a rigid analytic version of Hironaka's Local Flattening Theorem, and establish a framework in which to study rigid subanalytic sets in the style of Denef and van den Dries.

1991 Mathematics Subject Classification: 32P05, 32B20, 12J25, 03C10.

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an $N$-dimensional rectangle under Neumann boundary conditions.

1991 Mathematics Subject Classification: 58C27, 58F14.

A general selection theorem for a Vietoris-like hyperspace topology is obtained and, as a result, several known selection theorems for both `topology'- and `metric'-generated hyperspace topologies are improved from a common point of view. Applications are presented in characterizing classes of metrizable spaces by continuous selections for their hyperspaces.

1991 Mathematics Subject Classification: primary 54B20, 54C65; secondary 54E45, 54D65, 54E40, 54C10, 54G12.