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 We also comment on the relation of our work to some unpublished work of Bott and Tate.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 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 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.

1991 Mathematics Subject Classification: 14J10, 32J15.

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.

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.

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.

Strong operator modules over a von Neumann algebra $R$ are introduced and the so-called extended Haagerup tensor product over $R$ of strong modules is studied. In the case $R={\bf C}$ and the spaces are weak* closed this product agrees with the weak* Haagerup tensor product of Blecher and Smith. If $C$ is the center of $R$, the extended Haagerup tensor product \rcr\ is a Banach algebra containing the central tensor product $R\haagc R$ of Chatterjee and Smith and has very nice properties concerning slice maps and the relative commutants of subalgebras. It is shown that \rcr\ is a dual Banach space, and all weak* closed two-sided ideals of the algebra \rcr\ are determined.

1991 Mathematics Subject Classification: 46L05