Algebraic gamma monomials and double coverings of cyclotomic fields

We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex , to compute , where is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension , we define a double covering of to be an extension of degree , such that is Galois. We demonstrate that each class gives rise to a double covering of , by . When lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of is abelian and hence . The may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

     

Quantum -space

The prime and primitive spectra of , the multiparameter quantized coordinate ring of affine -space over an algebraically closed field , are shown to be topological quotients of the corresponding classical spectra, and , provided the multiplicative group generated by the entries of avoids .

     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

     

Fundamental groups of moduli and the Grothendieck-Teichmüller group

Let denote the moduli space of Riemann spheres with ordered marked points. In this article we define the group of quasi-special symmetric outer automorphisms of the algebraic fundamental group for all to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of and commuting with the group of outer automorphisms of obtained by permuting the marked points. Our main result states that is isomorphic to the Grothendieck-Teichmüller group for all .

     

The range of traces on quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product -algebra. This enables us to show that all tracial states on induce the same homomorphism on , whose range is the group .

     

On the stable module category of a self-injective algebra

Let be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable -modules which belong to Auslander-Reiten components of the form or . The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.     

Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation:

where and are two positive parameters and is a smooth bounded domain in containing in its interior. The variational approach requires that , and , which we assume throughout. However, the situations differ widely with and , and the interesting cases occur either at the critical Sobolev exponent () or in the Hardy-critical setting () or in the more general Hardy-Sobolev setting when . In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case , especially those corresponding to singularities (i.e., when .

     

A join theorem for the computably enumerable degrees

It is shown that for any computably enumerable (c.e.) degree , if , then there is a c.e. degree such that (so is lowand is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low c.e. degrees are not elementarily equivalent as partial orderings.

     

The zero set of semi-invariants for extended Dynkin quivers

We show that the set of common zeros of all semi-invariants vanishing at 0 on the variety of all representations with dimension vector of an extended Dynkin quiver under the group is a complete intersection if is ``big enough'. In case does not contain an open -orbit, which is the case not considered so far, the number of irreducible components of grows with , except if is an oriented cycle.

     

Mean convergence of orthogonal Fourier series of modified functions

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the -metric for any 2. $"> We prove also that if is a uniformly bounded ONS which is complete in all the spaces , then there exists a rearrangement of the natural numbers such that the system has the strong -property for all 2$">; that is, for every and for every and 0 $">there exists a function which coincides with except on a set of measure less than and whose Fourier series with respect to the system converges in      

Maximal holonomy of infra-nilmanifolds with -dimensional quaternionic Heisenberg geometry

Let be the quaternionic Heisenberg group of real dimension and let denote the maximal order of the holonomy groups of all infra-nilmanifolds with -geometry. We prove that . As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a -dimensional quaternionic hyperbolic manifold with cusps is at least

     

Arithmetic rigidity and units in group rings

For any finite group the group of units in the integral group ring is an arithmetic group in a reductive algebraic group, namely the Zariski closure of . In particular, the isomorphism type of the -algebra determines the commensurability class of ; we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the -representations of the converse is exactly true.

     

Specializations of Brauer classes over algebraic function fields

Let be either a number field or a field finitely generated of transcendence degree over a Hilbertian field of characteristic 0, let be the rational function field in one variable over , and let . It is known that there exist infinitely many such that the specialization induces a specialization , where has exponent equal to that of . Now let be a finite extension of and let . We give sufficient conditions on and for there to exist infinitely many such that the specialization has an extension to inducing a specialization , the residue field of , where has exponent equal to that of . We also give examples to show that, in general, such need not exist.

     

Semi-classical limit for random walks

Let be a discrete symmetric random walk on a compact Lie group with step distribution and let be the associated transition operator on . The irreducibles of the left regular representation of on are finite dimensional invariant subspaces for and the spectrum of is the union of the sub-spectra on the irreducibles, which consist of real eigenvalues . Our main result is an asymptotic expansion for the spectral measures

along rays of representations in a positive Weyl chamber , i.e. for sequences of representations , with . As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on (for which is essentially a direct sum of Harper operators).

     

The flat model structure on complexes of sheaves

Let be the category of chain complexes of -modules on a topological space (where is a sheaf of rings on ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on . As a corollary, we have a general framework for doing homological algebra in the category of -modules. I.e., we have a natural way to define the functors and in .

     

The geometry of profinite graphs with applications to free groups and finite monoids

We initiate the study of the class of profinite graphs defined by the following geometric property: for any two vertices and of , there is a (unique) smallest connected profinite subgraph of containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition.

We define a pseudovariety of groups to be arboreous if all finitely generated free pro- groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties , a pro- analog of the Ribes and Zalesski product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions to the much studied pseudovariety equation .

     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

Products and duality in Waldhausen categories

The natural transformation from -theory to the Tate cohomology of acting on -theory commutes with external products. Corollary: The Tate cohomology of acting on the -theory of any ring with involution is a generalized Eilenberg-Mac Lane spectrum, and it is 4-periodic.

     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.