For an $l\times k$ matrix $A=(a_{ij})$ of integers, denote by $\mathcal L(A)$ the system of homogenous linear equations $a_{i1}x_1+\ldots+a_{ik}x_k=0$, $1\le i\le l$. We say that $A$ is {\it density regular} if every subset of $\bold N$ with positive density, contains a solution to $\mathcal L(A)$. For a density regular $l\times k$ matrix $A$, an integer $r$ and a set of integers $F$, we write $$F\to(A)_r$$ if for any partition $F=F_1\cup....\cup F_r$ there exists $i\in\{1,2,...,r\}$ and a column vector $\bold x$ such that $A\bold x=\bold 0$ and all entries of $\bold x$ belong to $F_i$. Let $[n]_N$ be a random $N$-element subset of $\{1,2,...,n\}$ chosen uniformly from among all such subsets. In this paper we determine for every density regular matrix $A$ a parameter $\alpha=\alpha(A)$ such that $\lim_{n\to\infty}\bold P([n]_N\to(A)_r)=0$ if $N=O(n^{\alpha})$ and 1 if $N=\Omega(n^{\alpha})$.

1991 Mathematics Subject Classification: 05D10, 11B25, 60C05

In this paper, we prove that the modular curve $X(11)$ over a field of characteristic 3 admits the Mathieu group $M_{11}$ as an automorphism group. We also examine some aspects of the geometry of the curve $X(11)$ in characteristic 3. In particular, we show that every point of the curve is a point of inflection, the curve has 110 hyperflexes and there are no inflectional triangles and 11232 inflectional pentagons, of which 144 are self-conjugate. The hyperflexes correspond to the supersingular elliptic curves. We comment on the relationship of Ward's quadrilinear invariant for $M_{12}$ to our work and announce for the first time the equations for Klein's A-curve of level 11. We also comment on the relation of our work to some unpublished work of Bott and Tate.

1991 Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10, 14N10, 20B25, 20C34.

In this paper we prove an analogue of Jantzen's sum formula for the $q$-Weyl modules of the $q$-Schur algebra and, as a consequence, derive the analogue of Schaper's theorem for the $q$-Specht modules of the Hecke algebras of type $\bf A$. We apply these results to classify the irreducible $q$-Weyl modules and the irreducible ($e$-regular) $q$-Specht modules, defined over any field. In turn, this allows us to identify all of the ordinary irreducible representations of the finite general linear group $GL_n(q)$ which remain irreducible modulo a prime $p$ not dividing~$q$.

1991 Mathematics Subject Classification: 20C32.

Let $(W,S)$ be a finite Coxeter system and $B=B(W)$ the corresponding Artin-Tits braid group. The natural map $B \rightarrow W$ has a canonical section $r:W \rightarrow B$ defined by the condition that if $w \in W$ is written as a reduced expression in the generators in ~$S$ then $r(w)$ is the corresponding product taken in $B$. The main result of the present paper is as follows. Let $C$ be a conjugacy class in $W$ whose elements have order $d$ say. Then there exists an element $w \in C$ of minimal length in $C$ such that $r(w)^d$ is a product of terms of the form $r(w_I)^{d_I}$ where the following hold: $d_I$ is a non-negative even integer, $I$ runs over a sequence of subsets of $S$ which decreases (which implies that the terms commute), and $w_I$ is the longest element in the corresponding parabolic subgroup of $W$. Such an element will be called a ‘good’ element in $C$. The result is proved case by case, using the classification of irreducible finite Coxeter groups and the knowledge of representatives of minimal length from the article by Geck and Pfeiffer in Advances in Math. 102 (1993) 79-94.The main application of this result concerns the problem of calculating character values of Iwahori-Hecke algebras. The generic Iwahori-Hecke algebra $H$ associated with $(W,S)$ is a quotient of the group algebra of $B$ by the ideal generated by quadratic relations of the form $(s-q)(s+1)$ where $s \in S$ and $q$ is an indeterminate. Thus, $H$ is an algebra over a suitable field of rational functions in the variable ~$q$. The above result implies that if $w$ is a good element in the class $C$ of $W$, then the eigen values of the standard basis element $T_w$ of $H$ in an irreducible representation of $H$ are roots of unity times fractional powers of ~$q$, and the fractional powers occurring can be explicitly determined from the ordinary character table of $W$. This result is used to compute the character table of the Iwahori-Hecke algebra of type $E_8$. To determine the roots of unity, we use additional relations coming from the modular representation theory of ~$H$. This completes the program of determining the character tables of Iwahori-Hecke algebras.

1991 Mathematics Subject Classification: 20C20, 20F36.

The general theory of locally coherent Grothendieck categories is presented. To each locally coherent Grothendieck category $\C$ a topological space, the Ziegler spectrum of $\C,$ is associated. It is proved that the open subsets of the Ziegler spectrum of $\C$ are in bijective correspondence with the Serre subcategories of $\coh \C,$ the subcategory of coherent objects of $\C.$ This is a Nullstellensatz for locally coherent Grothendieck categories. If $R$ is a ring, there is a canonical locally coherent Grothendieck category $\RC$ (respectively, $\CR$) used for the study of left (respectively, right) $R$-modules. This category contains the category of $R$-modules and its Ziegler spectrum is quasi-compact, a property used to construct large (not finitely generated) indecomposable modules over an artin algebra. Two kinds of examples of locally coherent Grothendieck categories are given: the abstract category theoretic examples arising from torsion and localization and the examples that arise from particular modules over the ring $R.$ The duality between $\coh (\RC)$ and $\coh \CR$ is shown to give an isomorphism between the topologies of the left and right Ziegler spectra of a ring $R.$ The Nullstellensatz is used to give a proof of the result of Crawley-Boevey that every character $\xi: K_0 (\coh \C) \to Z$ is uniquely expressible as a $Z$-linear combination of irreducible characters.

1991 Mathematics Subject Classification: 16D90, 18E15.

In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions $N/K$ of quadratic imaginary fields $K$ with group $G$. For abelian groups $G$, the conjecture specialises to the statement that Chinburg's third invariant $\Omega(N/K,3)=0 \in \mbox{Cl}({\Bbb Z}[G])$. We prove (Theorem 1) that if $K$ has class number 1 (and some technical conditions on the conductor of $N/K$ and its roots of unity hold) then $\Omega(N/K,3)$ lies in the kernel group $D({\Bbb Z}[G])$. For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that $\Omega(N/K,3)=0$. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order ${\mathcal M}_G$ in${\Bbb Q}[G]$. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated ${\Bbb Z}[G]$-modules, which is isomorphic to $\mbox{Cl}({\mathcal M}_G)$. For Theorem 2, the fact that the class number of $N$ is coprime to $[N:K]$ in such genus extensions $N/K$ allows one to lift these results to $\mbox{Cl}({\Bbb Z}[G])$, after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining $\Omega (N/K,3)$ has the correct extension class (which is related to the fundamental class of $N/K$).

1991 Mathematics Subject Classification: 11R33.

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 construct quantum deformations of imaginary Verma modules over $U(A_1^{(1)})$ and show that, for generic $q$, imaginary Verma modules over $U(A_1^{(1)})$ can be deformed to those over the quantum group $U_q(A_1^{(1)})$ in such a way that the dimensions of the weight spaces are invariant under the deformation. We also prove the PBW theorem for $U_q(A_1^{(1)})$ with respect to the triangular decomposition induced from the root partition corresponding to the imaginary Verma modules.

1991 Mathematics Subject Classification: 17B67, 17B65, 17B10.

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

This paper gives one-term componentwise asymptotics for the $M$ and spectral matrices of a self-adjoint realisation of an even-order ordinary differential expression. The underlying interval is assumed to have at least one regular endpoint, and the boundary conditions are supposed to be separated. Furthermore, the weight function and the reciprocal of the highest-order coefficient are supposed to be of regular variation at the regular endpoint, in the sense of Bingham, Goldie and Teugels.

1991 Mathematics Subject Classification: 34B24, 34E05.

1991 Mathematics Subject Classification: 14J10, 32J15.

We study the subgroup growth of profinite groups. We obtain a structure theorem for profinite groups of polynomial subgroup growth (PSG groups, for short), which essentially reduces their characterization to the case where the group is a cartesian product of finite simple groups. Analysing the growth behaviour of such cartesian products, we construct, for any real number $\alpha \ge 1$, a PSG profinite group whose degree is exactly $\alpha$. Applications to the behaviour of the abscissa of convergence of the associated zeta function $\sum a_n(G)n^{-s}$ are drawn. We also show that there is no gap between polynomial and non-polynomial subgroup growth by constructing non-PSG groups whose subgroup growth is arbitrarily slow. Our arguments rely heavily on the use of sieve methods in number theory. In particular, a Bombieri-type short intervals theorem and the so-called Fundamental lemma in sieve theory play an essential role in this paper.

1991 Mathematics Subject Classification: 20E07, 11N36.

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial $f$ and an arithmetical invariant associated to a polynomial $f$ over a $p$-adic field.

When $f$ is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of $f^{-1} \{ 0 \}$ in $\Bbb A^2$. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘$q$-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity.

1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)

The isomorphism types of finite Lie primitive subgroups of the complex Lie groups $E_{6} ( \C )$ and $F_4 ( \C )$ are determined. Here, we call a finite subgroup of a complex Lie group $G$ {\sl Lie primitive\/} if it is not contained in a proper closed subgroup of $G$ of positive dimension. Induction can be used to investigate subgroups which are not Lie primitive. Some additional information is provided, such as the characters of these finite subgroups on some small-dimensional modules for the Lie groups.

In studying these groups, we mainly use two rational linear representations of the universal covering group $\widetilde E$ of $E_6(\C)$, namely a 27-dimensional module (there are two inequivalent ones), denoted by $\K$, and the adjoint module. In particular, we make heavy use of the characters of $\widetilde E$ on these modules. The group $F_4(\C)$ occurs in $\widetilde E$ as the stabilizer subgroup of a vector in $\K$. The finite simple groups of which a perfect central extension occurs in $F_4(\C)$ or $E_6(\C)$ are:

via $G_2$: $Alt_5$, $Alt_6$, $L(2,7)$, $L(2,8)$, $L(2,13)$, $U(3,3)$,

via $F_4$: $Alt_7$, $Alt_8$, $Alt_9$, $L(2,17)$, $L(2,25)$, $L(2,27)$, $L(3,3)$, ${}^3D_4(2)$, $U(4,2)$, $O(7,2)$, $O^{+}(8,2)$,

via $E_6$: $Alt_{10}$, $Alt_{11}$, $L ( 2 , 11)$, $L(2,19)$, $L(3,4)$, $U(4,3)$, ${}^2F_4(2)'$, $M_{11}$, $J_2$.

This list has been found using the classification of the finite simple groups. On the basis of this list, the finite Lie primitive subgroups are found to be either the normalizers of one of these subgroups or of one of the two elementary abelian $3$-groups found by Alekseevskii.

1991 Mathematics Subject Classification: 20K47, 20G40, 17B45, 20C10.

Semisimple Banach algebras are well-known to have a unique (complete) algebra norm topology, but such uniqueness may fail if the radical is even one-dimensional. We obtain a necessary condition for uniqueness of norm when the algebra has finite-dimensional radical. In the case where the Banach algebra is separable, the condition is shown to be also sufficient for a large class of algebras, and in particular under various hypotheses of commutativity. Examples are given to show the limitations of the various sufficiency results, and these also give a good indication of where the difficulties lie in general. We conjecture that, at least in the separable case, our condition is both necessary and sufficient.

1991 Mathematics Subject Classification: 46H20, 46H40.

Let $f$, $g$ : $({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$ be two $C^\infty$ map-germs. Then $f$ and $g$ are $C^0$-equivalent if there exist homeomorphism-germs $h$ and $l$ of $({\Bbb R}^n,0)$ and $({\Bbb R}^p,0)$ respectively such that $g=l\circ f\circ h^{-1}$. Let $k$ be a positive integer. A germ $f$ is $k$-$C^0$-determined if every germ $g$ with $j^k g(0)=j^k f(0)$ is $C^0$-equivalent to $f$. Moreover, we say that $f$ is finitely topologically determined if $f$ is $k$-$C^0$-determined for some finite $k$. We prove a theorem giving a sufficient condition for a germ to be finitely topologically determined. We explain this condition below.

Let $N$ and $P$ be two $C^\infty$ manifolds. Consider the jet bundle $J^k(N,P)$ with fiber $J^k(n,p)$. Let $z\in J^k(n,p)$ and let $f$ be such that $z= j^kf(0)$. Define \[ \chi(f)=\dim_{\Bbb R} \frac{\theta (f)} {tf(\theta (n))+f^{\ast}(m_p)\theta(f)}. \] Whether $\chi(f)(k$ depends only on $z$, not on $f$ . We can therefore define the set $W^k= W^k(n,p)=\{z\in J^k(n,p)\vert \chi(f)\ge k$ for some representative $f$ of $z$\}. Let $W^k(N,P)$ be the subbundle of $J^k(N,P)$ with fiber $W^k(n,p)$. Mather has constructed a finite Whitney (b)-regular stratification ${\mathcal{S}}^k(n,p)$ of $J^k(n,p)-W^k(n,p)$ such that all strata are semialgebraic and $\mathcal K$-invariant, having the property that if ${\mathcal S}^k(N,P)$ denotes the corresponding stratification of $J^k(N,P)-W^k(N,P)$ and $f\in C^\infty (N,P)$ is a $C^\infty$ map such that $j^k f$ is multitransverse to ${\mathcal S}^k(N,P)$, $j^k f(N)\cap W^k(N,P)=\emptyset$ and $N$ is compact (or $f$ is proper), then $f$ is topologically stable.

For a map-germ $f:({\Bbb R}^n,0)\rightarrow ({\Bbb R}^p,0)$, we define a certain {\L}ojasiewicz inequality. The inequality implies that there exists a representative $f : U\rightarrow{\Bbb R}^p$ such that $j^kf(U-\{0\})\cap W^k({\Bbb R}^n,{\Bbb R}^p)=\emptyset$ and such that $j^kf$ is multitransverse to ${\mathcal S}^k({\Bbb R}^n,{\Bbb R}^p)$ at any finite set of points $S\subset U-\{0\}$. Moreover, the inequality controls the rate $j^k f$ becomes non-transverse as we approach 0. We show that if $f$ satisfies this inequality, then $f$ is finitely topologically determined.

1991 Mathematics Subject Classification: 58C27.

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.

The Bieri-Neumann-Strebel invariant of a finitely generated group $G$ determines, among other things, whether or not a given normal subgroup $N$, with $G/N$ abelian, is finitely generated. We examine the BNS-invariants of “Pride groups”, a large class of groups containing the Artin groups; in particular we establish a criterion which implies that a character $\chi$ of a Pride group $G{\cal G}$ is in the BNS-invariant $\Sigma^1(G{\cal G})$.

This restricts in the case of an Artin group $A{\cal G}$ to a simple condition which implies a character is in $\Sigma^1(A{\cal G})$. As an application this can be used to compute the unit ball in the Thurston semi-norm for the complement of iterated connected sums of $(2,n)$-torus links.

1991 Mathematics Subject Classification: 20F36, 20E07.

Let $\Omega$ be an open proper subset of $\Bbb R^n$. Its {\it skeleton} is the set of points with more than one nearest neighbour in the complement of $\Omega$; its {\it central set} is the set of centres in maximal open balls included in $\Omega$. Intuitively, if we think of $\Omega$ as a land mass in which height is proportional to distance from the sea, its skeleton and central set can be thought of as corresponding to ridges in the mountains of $\Omega$. In this note I discuss the metric and topological properties of such sets. I show that any skeleton in $\Bbb R^n$ is F$_{\sigma}$, and has dimension at most $n-1$, by any of the usual measures of dimension; that if $\Omega$ is bounded and connected, its skeleton and central set are connected; and that $\Omega$ separates $\Bbb R^n$ iff its skeleton does iff its central set does. Any central set in $\Bbb R^n$ is a G$_{\delta}$ set of topological dimension at most $n-1$. In the plane, I show that both skeletons and central sets are locally path-connected, and indeed include many paths of finite length. For any $\Omega$, its central set includes its skeleton; I give examples to show that the central set can be significantly larger than the skeleton.

1991 Mathematics Subject Classification: 54F99.

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.