Smooth projective varieties with small invariants have received renewed interest in recent years, primarily due to the fine study of the adjunction mapping. Now, through the effort of several mathematicians, a complete classification of smooth surfaces in ${\Bbb P}^4$ has been worked out up to degree $10$, and a partial one is available in degree $11$. On the other side, recently Ellingsrud and Peskine have proved Hartshorne's conjecture that there are only finitely many families of smooth surfaces in ${\Bbb P}^4$, not of general type. It is believed that the degree of the smooth, non-general type surfaces in ${\Bbb P}^4$ should be less than or equal to $15$.

The aim of this paper is to provide a series of examples of smooth surfaces in ${\Bbb P}^4$, not of general type, in degrees varying from $12$ up to $14$, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct the following families of surfaces: \begin{enumerate} \item[] minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=13$; \item[] two types of non-minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=14$; \item[] non-minimal $K3$ surfaces of degree $13$ and sectional genus $16$; and \item[] non-minimal $K3$ surfaces of degree $14$ and sectional genus $19$.

1991 Mathematics Subject Classification: 14M07, 14J25, 14J26, 14J28, 14C05.

We investigate the structure of transitive, untwisted, superlinked finite covers whose kernels are central in their automorphism groups. We introduce the concept of an $n$-conjugate system for a pair $(W,K)$, where $W$ is a permutation structure, $K$ is a finite abelian group and $n\in\omega+1$. This concept allows us to characterize the given class of finite covers for structures $W$ which satisfy a certain connectedness condition; further, the irreducibility of such a cover is equivalent to a simple condition on a corresponding $n$-conjugate system. Finally, we consider structures with strong types, for which there is a much simpler characterization.

1991 Mathematics Subject Classification: 03C35, 20B27.

Assuming the Riemann Hypothesis, Montgomery showed by means of his pair correlation method that at least two-thirds of the zeros of Riemann's zeta-function are simple. Later he and Taylor improved this to 67.25 percent and, more recently, Cheer and Goldston increased the percentage to 67.2753. Here we prove by a new method that if the Riemann and Generalized Lindel\"of Hypotheses hold, then at least 70.3704 percent of the zeros are simple and at least 84.5679 percent are distinct. Our method uses mean value estimates for various functions defined by Dirichlet series sampled at the zeros of the Riemann zeta-function.

1991 Mathematics Subject Classification: 11M26.

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.

Let $p_n$ be the $n$th prime. Then this paper is concerned with proving the following result on the distribution of consecutive primes.

Theorem.}\begin{equation}\sum_{p_{n+1}-p_n>x^{\frac 12},\ x \leq p_n \leq 2x} (p_{n+1}-p_n) \llx^{\frac{25}{36}+\epsilon}.\end{equation}

The exponent of $x$ in this theorem improves on the work of Heath-Brown who proved $(1)$ with exponent $\frac 34$. Under the Riemann hypothesis one can prove$(1)$ with exponent $\frac 12$.The proof of the theorem starts with the Heath-Brown--Linnik identity which leads to a formula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. I then estimate the coefficients by using, among other things, the information which can be gained from Montgomery's mean value theorem and Huxley's version of the Hal\' asz lemma. Furthermore, by using familiar sieve arguments I am able to discard some of the coefficients allowing us to gain an improvement over the previous result of Heath-Brown.

1991 Mathematics Subject Classification: 11N05.

Suppose a group $G$ acts on a Gromov-hyperbolic space $X$ properly discontinuously. If the limit set $L(G)$ of the action has at least three points, then the second bounded cohomology group of $G$,$H^2_b(G;{\Bbb R})$ is infinite dimensional. For example, if $M$ is a complete, pinched negatively curved Riemannian manifold with finite volume, then $H_b^2(\pi _1(M); {\Bbb R})$ is infinite dimensional. As an application, we show that if $G$ is a knot group with $G \not\simeq{\Bbb Z}$, then $H^2_b(G;{\Bbb R})$ is infinite dimensional.

1991 Mathematics Subject Classification: primary 20F32; secondary 53C20, 57M25.

A Mackey functor $M$ is a structure analogous to the representation ring functor $H \mapsto R(H)$ encoding good formal behaviour under induction and restriction. More explicitly, $M$ associates an abelian group $M(H)$ to each closed subgroup $H$ of a fixed compact Lie group $G$, and to each inclusion $K \subseteq H$ it associates a restriction map ${\rm res}^H_K:M(H) \rightarrow M(K)$ and an induction map${\rm ind}^H_K:M(K) \rightarrow M(H)$. This paper gives an analysis of the category of Mackey functors $M$ whose values are rational vector spaces: such a Mackey functor may be specified by giving a suitably continuous family consisting of a ${\Bbb Q} \pi_0(W_G(H))$-module $V(H)$ for each closed subgroup $H$ with restriction maps $V(\hat{K}) \rightarrow V(K)$ whenever $K$ is normal in $\hat{K}$ and $\hat{K}/K$ is a torus (a ‘continuous Weyl-toral module’). We show that the category of rational Mackey functors is equivalent to the category of rational continuous Weyl-toral modules. In Part II this will be used to give an algebraic analysis of the category of rational Mackey functors, showing in particular that it has homological dimension equal to the rank of the group.

1991 Mathematics Subject Classification: 19A22, 20C99, 22E15, 55N91, 55P42, 55P91.

The leading asymptotic term for the function that counts the eigenvalues of the Stokes operator is determined for fairly general underlying bounded domains. Moreover, the remainder is estimated in terms of the fractality of the boundary of the domain. The results obtained resemble corresponding ones for the Dirichlet Laplacian.

1991 Mathematics Subject Classification: 35P20.

This paper is concerned with rational representations of reductive algebraic groups over fields of positive characteristic $p$. Let $G$ be a simple algebraic group of rank $\ell$. It is shown that a rational representation of $G$ is semisimple provided that its dimension does not exceed $\ell p$. Furthermore, this result is improved by introducing a certain quantity $\mathcal{C}$ which is a quadratic function of $\ell$. Roughly speaking, it is shown that any rational $G$ module of dimension less than $\mathcal{C} p$ is either semisimple or involves a subquotient from a finite list of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. The essential problem is to study the possible extensions between $L_1$ and $L_2$ provided $\dim L_1 + \dim L_2$ is smaller than $\mathcal{C} p$. In this paper, all relevant simple modules $L_i$ are characterized, the restricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of the corresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of the extension theory.

1991 Mathematics Subject Classification: 20G05.

A notion of $L^2$-determinant is established generalizing $L^2$-torsion. It is shown that classical regularized determinants of elliptic operators converge to $L^2$-determinants in a tower of coverings. A product formula is proved, which provides an Euler-product expansion to regularized determinants, having generalized $L^2$-determinants as Euler-factors.

1991 Mathematics Subject Classification: 58G26, 58F20.

In this paper Hill's equation $y''+qy=Ey$, where $q$ is a complex-valued function with inverse square singularities, is studied. Results on the dependence of solutions to initial value problems on the parameter $E$ and the initial point $x_0$, on the structure of the conditional stability set, and on the asymptotic distribution of (semi-)periodic and Sturm-Liouville eigenvalues are obtained. It is proved that a certain subset of the set of Floquet solutions is a line bundle on a certain analytic curve in ${\Bbb C}^2$. We establish necessary and sufficient conditions for $q$ to be algebro-geometric, that is, to be a stationary solution of some equation in the Korteweg-de Vries (KdV) hierarchy. To do this a distinction between movable and immovable Dirichlet eigenvalues is employed. Finally, an example showing that the finite-band property does not imply that $q$ is algebro-geometric is given. This is in contrast to the case where $q$ is real and non-singular.

1991 Mathematics Subject Classification: 34L40, 14H60.

Much is known about the connection between the growth and decay of subharmonic functions. The results indicate that there is a general principle: {\em asubharmonic function cannot decay ‘too fast’ relative to its growth}.Three theorems are proved which, together with work previously published elsewhere, give a fairly complete account of how this principle works out for a subharmonic function having extremal decay along a ray.

1991 Mathematics Subject Classification: 30D20, 31A05.

A function f : Rn → R is a connectivity function if the graph of its restriction f|C to any connected C ⊂ Rn is connected in Rn x R. The main goal of this paper is to prove that every function f : Rn → R is a sum of n + 1 connectivity functions (Corollary 2.2). We will also show that if n > 1, then every function g: Rn → R which is a sum of n connectivity functions is continuous on some perfect set (see Theorem 2.5) which implies that the number n + 1 in our theorem is best possible (Corollary 2.6).

To prove the above results, we establish and then apply the following theorems which are of interest on their own.

For every dense Gδ-subset G of Rn there are homeomorphisms h1...,hn of Rn such that Rn=G∪h1(G)∪...∪hn(G) (Proposition 2.4).

For every n >1 and any connectivity function f : Rn → R, if x ∈ Rn and ε >0 then there exists an open set U⊂Rn such that x ∈ U⊂Bn(x,ε), f|bd(U) is continuous, and |(x)-f(y)| < ε for every y∈bd(U) (Proposition 2.7).

We give a complete classification (up to unitary equivalence) of extensions of $C(X)$ by a separable simple AF-algebra $A$ with a unique trace (up to scalar multiples), where $X$ is a compact subset of the plane. In particular, we show that there are non-trivial extensions $\tau$ such that $[\tau]=0$ in ${\rm Ext}(C(X),A).$ A new index is introduced to determine when an extension is trivial. Extensions of $C(S^2)$ and other algebras are also studied. Our results work for a larger class of C*-algebras of real rank zero.

1991 Mathematics Subject Classification: primary 46L05,46L35; secondary 46L80.

Given a generalized octagon which satisfies certain geometric conditions, we show that it must be isomorphic to the Ree--Tits octagon arisingnaturally from a Ree group of type $^2F_4$ over a perfect field of characteristic 2, in which the Frobenius automorphism has a square root.

1991 Mathematics Subject Classification: 51E12.