The aim of the paper is to construct a calculus of pseudodifferential operators (PDOs) on a smooth manifold $M$ without using local coordinate systems. Instead we deal with linear connections $\Gamma$ of $M$.

The fact that a linear connection $\Gamma$ is a global object enables one to associate with a PDO its full symbol, which is a function on the cotangent bundle $T^*M$ (depending on the choice of $\Gamma\,$). This idea was put forward by H. Widom. He has suggested a method of defining full symbols of PDOs on a manifold with a linear connection and constructed a version of symbolic calculus. However, H. Widom defined the classes of PDOs in local coordinates, using standard local phase functions.

On the contrary, in the paper the PDOs are defined in a coordinate-free way, with use of invariant oscillatory integrals over $T^*M\,$. The invariant approach allows us to define $\tau$-symbols of pseudodifferential operators acting on a manifold; in particular, we introduce the Weyl symbols. In fact, it is a kind of quantization on a manifold provided with a linear connection.

The results are applied to study of semi-elliptic differential operators and functions of the Laplacian on a closed Riemannian manifold.

1991 Mathematical Subject Classification: 35S05, 58G30.

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical L-functions of arithmetic orders considered by Hey, Solomon, Bushnell and Reiner. We calculate the zeta functions and normal zeta functions of the plane crystallographic groups. As a corollary of these calculations we produce \begin{enumerate} \item[(1)] examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and \item[(2)] examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product.

1991 Mathematics Subject Classification: 11M41, 20H15.

Let $L(V)$ be the free Lie algebra on a finite-dimensional vector space $V$ over a field $K$, with homogeneous components $L^n(V)$ for $n \geq 1$. If $G$ is a group and $V$ is a $KG$-module, the action of $G$ extends naturally to $L(V)$, and the $L^n(V)$ become finite-dimensional $KG$-modules, called the Lie powers of $V$. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable $KG$-modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where $G$ has prime order $p$, and $K$ has characteristic $p$. As is well known, there are $p$ indecomposables, denoted here by $J_1,\dots,J_p$, where $J_r$ has dimension $r$. A theory is developed which provides information about the overall module structure of $L(V)$ and gives a recursive method for finding the multiplicities of $J_1,\dots,J_p$ in the Lie powers $L^n(V)$. For example, the theory yields decompositions of $L(V)$ as a direct sum of modules isomorphic either to $J_1$ or to an infinite sum of the form $J_r \oplus J_{p-1} \oplus J_{p-1} \oplus \ldots $ with $r \geq 2$. Closed formulae are obtained for the multiplicities of $J_1,\dots,J_p$ in $L^n(J_p)$ and $L^n(J_{p-1})$. For $r < p-1$, the indecomposables which occur with non-zero multiplicity in $L^n(J_r)$ are identified for all sufficiently large $n$.

2000 Mathematical Subject Classification: 17B01, 20C20.

The paper considers pairs $(X,B)$ where $X$ is a normal projective surface over $\Bbb C$, and $B$ is a $\Bbb Q$-divisor whose coefficients are $1$ or $1-1/m$ for some natural number $m$. A log canonical singularity on such a pair is a quotient by a finite or infinite group, so if $(X,B)$ has log canonical singularities, the orbifold Euler number $e_{\rm orb}(X,B)$ can be defined. The main result is a Bogomolov-Miyaoka-Yau-type inequality which implies that if $(X,B)$ has log canonical singularities and $\kappa(X,K_X+B)\ge 0$ then $(K_X+B)^2\le 3e_{\rm orb}(X,B)$. The actual inequality proved is somewhat stronger and it also implies all the previously published versions of the Bogomolov-Miyaoka-Yau inequality. The proof involves the Log Minimal Model Program, $Q$-sheaves when $K_X+B$ is nef, and a study of the changes in the two sides of the inequality under a contraction. The paper also contains a further generalisation where the coefficients of $B$ can be arbitrary rational numbers in $[0,1]$, a different condition is imposed on the singularities and $K_X+B$ is required to be nef. Some applications of the inequalities are also given, for example, estimating the number of singularities or certain kinds of configurations of curves on surfaces.

We show that if $G$ is a group of exponent 5, and if $G$ satisfies the Engel-4 identity $[x,y,y,y,y]=1$, then $G$ is locally finite. By a result of Traustason, this implies that Engel-4 5-groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity

$$[x,[y,z,z,z,z],[y,z,z,z,z]]=1.$$

This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite.

1991 Mathematics Subject Classification: 20D15, 20F45, 20F50.

We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a self-averaging integrated density of states and a Pastur–šubin type trace formula.

Dedicated to the memory of Professor A. I. Kostrikin

The main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations $\phi$ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations $\phi$ that can be realized over an unramified extension of ${\mathbb Q}_p$, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations $\rho$ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and $G\neq G_2(5)$, it is proved that either the Jordan normal form of a root element contains a block of size d with 1<d<p -1 or the highest weight of $\rho$ is equal to p -1 times the sum of the fundamental weights.

2000 Mathematical Subject Classification: 20C33, 20G15.

Let $F$ be an algebraically closed field of characteristic $p \geq 0$ and $A_n$ be the alternating group on $n$ letters. The main goal of this paper is to describe the pairs $(G, E)$, where $E$ is an irreducible $FA_n$-module and $G < A_n$ is a proper subgroup of $A_n$ such that the restriction $E{\downarrow_G}$ is irreducible. We are able to give a list of all such pairs, provided $p > 3$. The case $p = 0$ has been treated by J. Saxl. The problem is important for the classification of maximal subgroups in finite classical groups.

2000 Mathematical Subject Classification: 20C20, 20C30, 20B35, 20B20.

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$ regarded as an $FG$-submodule of the free associative algebra (or tensor algebra) $T(V)$. For each positive integer $r$, let $L^r (V)$ and $T^r (V)$ be the $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th Lie power of $V$. Our main result is that there are submodules $B_1$, $B_2$, ... of $L(V)$ such that, for all $r$, $B_r$ is a direct summand of $T^r(V)$ and, whenever $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. Thus every Lie power is a direct sum of Lie powers of $p$-power degree. The approach builds on an analysis of $T^r (V)$ as a bimodule for $G$ and the Solomon descent algebra.

In this paper a practical algorithm is given to find all binary forms with rational coefficients of given degree with discriminant divisible by a given finite set of rational primes, up to an obvious equivalence relation.

This is done by adapting the finiteness result of Evertse and Gy\H{o}ry. A technical assumption that all fields used have class number $1$ is made to aid the exposition.

All binary forms of degree less than or equal to $6$ with $2$-power discriminant are then calculated. This is then used to complete the table of curves of genus $2$ which had previously been computed by Merriman and the author. Some ranks of the associated Jacobian varieties are also computed.

1991 Mathematics Subject Classification: primary 11E76, 11G30; secondary 11Y40, 11D61.

Let $\varGamma =\varGamma _{\tau ,z}$ be one of the $N^2$-dimensional bicovariant first order differential calculi for the quantum groups $\mathrm{GL}_q(N)$, $\mathrm{SL}_q(N)$, $\mathrm{SO}_q(N)$, or $\mathrm{Sp}_q(N)$, where $q$ is a transcendental complex number and $z$ is a regular parameter. It is shown that the de Rham cohomology of Woronowicz' external algebra $\varGamma^\land $ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (two-sided) coinvariant subcomplexes. In the cases $\mathrm{GL}_q(N)$ and $\mathrm{SL}_q(N)$ the cohomology ring is isomorphic to the coinvariant external algebra $\varGamma ^\land _{\scriptscriptstyle{\mathrm{Inv}}}$ and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. 2000 Mathematical Subject Classification: 46L87, 58A12, 81R50.

We prove that abelian subgroups of fundamental groups of Haken 3-manifolds and closed hyperbolic n-orbifolds are separable. 2000 Mathematical Subject Classification: 20E26, 51M10, 57M05.

Representations of Hecke and $q$-Schur algebras are closely related to those of finite general linear groups $G$ in non-describing characteristics. Such a relationship can be described by certain functors. Using these functors, we determine the Harish-Chandra vertices and sources of certain indecomposable $G$-modules. The Green correspondence is investigated in this context. As a further application of our theory, we establish Steinberg's tensor product theorems for irreducible representations of $G$ in non-describing characteristics.

1991 Mathematics Subject Classification: 20C20, 20C33, 20G05, 20G40.

We study a $q$-deformation for the crossed product of the symmetric group $S_n$ with the Clifford algebra on $n$ anticommuting generators. We obtain a $q$-version of the projective analogue for the classical Young symmetrizer introduced by the second author. Our main tool is an analogue of the Hecke algebra of complex-valued functions on the group $\mbox{GL}_n$ over a $p$-adic field relative to the Iwahori subgroup.

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds which may embed as topologically locally flat submanifolds of $S^4$. In particular, a circle bundle over a non-orientable surface $F$ embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of $F$ in $S^4$, and we show that precisely thirteen geometric 3-manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0-framed surgery on a link which is the union of two slice links, and so embeds smoothly in $S^4$.

1991 Mathematics Subject Classification: 57N13.

We prove that, for every bounded and measurable forcing $p(t)$, the differential equation $\ddot{u}+u^{1/3} =p(t)$ has bounded solutions with arbitrarily large amplitude. In general it is not possible to say that all solutions are bounded, as shown by an example due to Littlewood.

The proof is based on a variational method which can be seen as a dual version of Nehari's method for boundary value problems on compact intervals.

In this paper a $T(1)$ theorem for Calder\'on--Zygmund operators (CZOs) valid for non-doubling measures with atoms is proved. In the classical Calder\'on--Zigmund theory an essential assumption is the doubling property of the underlying measure $\mu$ in ${\mathbb R}^n$. Some recent results have shown that a big part of the classical theory can be extended to the case of non-doubling measures, with only some mild `growth' condition on the measure $\mu$. In the particular case of the Cauchy transform $Cf(x) = \int f(y)/(y-x) \, d\mu(y)$, the main result of the paper can be stated easily: the Cauchy transform is bounded in $L^2(\mu)$ if and only if it is bounded in $L^2(\mu)$ over characteristic functions of squares. The proof of the main result is based on the use of the Haar basis with random dyadic lattices, following an idea of Nazarov, Treil and Volberg (`Cauchy integral and Calder\'on--Zygmund operators on nonhomogeneous spaces', {\em Internat. Math. Res. Notices} (1997) 15, 703--726), where a $T(1)$ theorem for non-doubling continuous measures was obtained. The case of a measure $\mu$ with atoms studied in the present paper is quite different from the case of continuous measures. For example, if $\mu$ has atoms, the $L^2(\mu)$ boundedness of a CZO does not imply its $L^p(\mu)$ boundedness for some $p\neq2$. The differences arise because a measure $\mu$ with atoms does not satisfy the essential `growth' condition that is involved in the arguments for proving the $L^2$ boundedness of CZOs for continuous measures. 2000 Mathematics Subject Classification: 42B20.

Let $G$ be a finite group.It is proved that if the probability that two randomly chosen elements of $G$ generate a soluble group is greater than $\frac{11}{30}$ then $G$ itself is soluble.The bound is sharp, since two elements of the alternating group $A_5$ generate $A_5$ withprobability $\frac{11}{30}$.Similar probabilistic statements are proved concerning nilpotency and the property of having odd order.It is also proved that there is a number $\kappa$,strictly between $0$ and $1$, with the following property. Let $\cal X$ be any class of finite groups which is closed for subgroups, quotient groups and extensions.If the probability that two randomly chosen elementsof $G$ generate a group in $\cal X$ is greater than $\kappa$ then $G$ is in $\cal X$. The proofs use the classification of the finite simple groups and also some of the detailed information now available concerning maximal subgroups of finite almost simple groups. 1991 Mathematics Subject Classification:20F16, 20D06, 20D08, 60B99.

Let $(S, 0)$ be a normal surface germ and Let $f$ a non-constant regular function on Let $(S, 0)$ with Let $f(0) = 0$. Using any additive invariant on complex algebraic varieties one can associate a zeta function to these data, where the topological and motivic zeta functions are the roughest and the finest zeta functions, respectively. In this paper we are interested in a geometric determination of the poles of these functions. The second author has already provided such a determination for the topological zeta function in the case of non-singular surfaces. Here we give a complete answer for all normal surfaces, at least on the motivic level. The topological zeta function however seems to be too rough for this purpose, although for negative poles, which are the only ones in the non-singular case, we are able to prove exactly the same result as for non-singular surfaces.

We also give and verify a (natural) definition for when a rational number is a pole of the motivic zeta function.

Our results are of three types. First, we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of $MU$, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable $K(n)$-localizations of such spectra. This generalizes the famous Hopkins–Miller theorem and gives strengthened versions of various splitting theorems.