Bulletin of the London Mathematical Society - Advance Access

Counting the discrete series for GL(n)

Bushnell, C. J., Henniart, G. — 2006-01-01

Let F be a non-Archimedean local field. Let n ≥ 1, j ≥ 0 be integers. This paper gives an exact formula for the number of equivalence classes of irreducible smooth representations of GL_n(F) which are square-integrable mod centre, satisfy a certain condition on the central character, and admit a fixed vector for the jth principal congruence subgroup of the maximal compact subgroup GL_n(o_F).

Trees, Gateaux norms and a problem of Haydon

Smith, R. J. — 2006-01-01

We give a sufficient condition for the existence of an equivalent Gâteaux smooth norm on C₀, where is a tree. Using this, we prove that if C₀ admits an equivalent strictly convex norm, then it admits an equivalent Gâteaux smooth norm. This resolves an open problem from Haydon's study of trees in renorming theory.

All tilting modules are of countable type

Stovicek, J., Trlifaj, J. — 2006-01-01

Let R be a ring and T an (infinitely generated) tilting module. Then T is of countable type; that is, there is a set, C, of modules possessing a projective resolution consisting of countably generated projective modules such that the tilting class T equals C. Moreover, a cotorsion pair C=(A, B) is tilting if and only if: C is hereditary, all modules in A have finite projective dimension, and B is closed under arbitrary direct sums.

Covolumes of uniform lattices acting on polyhedral complexes

Thomas, A. — 2006-01-01

Let X be a polyhedral complex with finitely many isometry classes of links. We establish a restriction on the covolumes of uniform lattices acting on X. When X is two-dimensional and has all links isometric to either a complete bipartite graph or the building for a Chevalley group of rank 2 over a field of prime order, we obtain further restrictions on covolumes.

Exotic smooth structures on 3CP2#8Formula

Park, J. — 2006-01-01

Motivated by Stipsicz and Szabó's exotic 4-manifolds with b₂⁺=3 and b₂^–=8, we construct a family of simply connected smooth 4-manifolds with b₂⁺=3 and b₂^–=8. As a corollary, we conclude that the topological 4-manifold 3CP²#8CP² admits infinitely many distinct smooth structures.

Gaussian bounds of heat kernels for Schrodinger operators on Riemannian manifolds

Takeda, M. — 2006-01-01

Suppose that the heat kernel on a complete Riemannian manifold satisfies global Gaussian bounds. We consider a Schrödinger operator for which the potential is a signed measure in a certain Kato class, and we establish a necessary and sufficient condition that the heat kernel of the Schrödinger operator also possesses the global Gaussian bounds.

Extremal metrics and K-stability

Szekelyhidi, G. — 2006-01-01

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson, which relate to the case of Kähler–Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

Entire cyclic homology of stable continuous trace algebras

Mathai, V., Stevenson, D. — 2006-01-01

A central result in this paper is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M.

An algebraic loop theorem and the decomposition of PD3-pairs

Crisp, J. — 2006-01-01

Let (Y, X) denote a three-dimensional Poincaré pair (PD³-pair). By the work of Eckmann, Müller and Linnell we may suppose, up to a homotopy equivalence, that the boundary X is a closed 2-manifold. We show that if a component of X fails to be ₁-injective in Y, then there is an essential simple loop in X which is nullhomotopic in Y. It follows that there is a finite process of attaching 2-disks along essential simple loops on X, and filling spherical components of X, which transforms (Y, X) into a PD³-pair (Y', X') with aspherical incompressible boundary X' and such that ₁(Y)=₁(Y'). The PD³-pair (Y', X') then admits a canonical decomposition as a connected sum of a finite number of aspherical PD³-pairs with incompressible boundary, together with a PD³-pair having virtually free (possibly finite) fundamental group and boundary a (possibly empty) disjoint union of projective planes.

Tameness and complexity of finite group schemes

Farnsteiner, R. — 2006-01-01

Using a representation-theoretic interpretation of support varieties due to Friedlander and Pevtsova (Amer. J. Math. 127 (2005) 379–420; Erratum Amer. J. Math. 128 (2006) 1067–1068, we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard (Bull. London Math. Soc. 22 (1990) 540–546), the proof of which is flawed.

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Lang, M. L. — 2006-01-01

Let =2cos (/5) and let Z[]. Denote the normaliser of G₀() of the Hecke group G₅ in PSL₂(R) by N(G₀()). Then N(G₀())=G₀(/h), where h is the largest divisor of 4 such that h² divides . Further, N(G₀())/G₀() is either 1 (if h=1), Z₂xZ₂ (if h=2) or Z₄xZ₄ (if h=4).

An Ekholm-Szucs-type formula for codimension one immersions of 3-manifolds up to bordism

Takase, M. — 2006-01-01

We give a formula for the bordism class of an immersion of an oriented 3-manifold in 4-space. It expresses the class in terms of the topology of a null-cobordism of the 3-manifold and certain singularities (the number of umbilic points) of a generic map of this null-cobordism into 4-space which extends the immersion.

Boundaries of reduced free group C*-algebras

Ozawa, N. — 2006-01-01

We prove that the crossed product C*-algebra C^*_r (, ) of a free group with its boundary sits naturally between the reduced group C*-algebra C^*_r and its injective envelope I(C^*_r ). In other words, we have natural inclusion C^*_r C^*_r (, )I(C^*_r ) of C*-algebras.

On a local-global principle for the divisibility of a rational point by a positive integer

Dvornicich, R., Zannier, U. — 2006-01-01

Following two previous papers (R. Dvornicich and U. Zannier, Bull. Soc. Math. France 129 (2001), 317–338; C. R. Acad. Sci. Paris, Ser. I 338 (2004) 47–50), we continue the investigation of a local-global principle for the divisibility by a positive integer of a rational point on a commutative algebraic group. In the first half of this paper some new affirmative results are obtained for elliptic curves. In the second half we investigate the structure of possible situations when the principle does not hold; it is shown that whenever a certain abstract cohomology group does not vanish (which ‘often’ happens) there exist negative examples over suitable number fields.