The aim of the present paper is the computation of Green's functions for the powers ${{\mathbf{\Delta }}^{m}}$ of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in ${{\mathbb{C}}^{d}}$, we obtain a complete result for $m\,=\,1,\,2$ in all dimensions. For $m\,\ge \,3$ the formulas grow quite complicated so we restrict ourselves to the case of the unit disc $(d\,=\,1)$ where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space ${{\mathbb{P}}^{d}}$ (for $d\,=\,1$, the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre $(d\,=\,1)$ or Jacobi $(d\,>\,1)$ polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed.

Let $N/K$ be a biquadratic extension of algebraic number fields, and $G\,=\,\text{Gal(}N/K\text{)}$. Under a weak restriction on the ramification filtration associated with each prime of $K$ above 2, we explicitly describe the $\mathbb{Z}\text{ }[G]\text{ }$-module structure of each ambiguous ideal of $N$. We find under this restriction that in the representation of each ambiguous ideal as a $\mathbb{Z}\text{ }[G]\text{ }$-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone.

For a given group, $G$, define ${{S}_{G}}$ to be the set of indecomposable $\mathbb{Z}\text{ }[G]\text{ }$-modules, $M$, such that there is an extension, $N/K$, for which $G\cong \text{Gal(}N/K\text{)}$, and $M$ is a $\mathbb{Z}\text{ }[G]\text{ }$-module summand of an ambiguous ideal of $N$. Can ${{S}_{G}}$ ever be infinite? In this paper we answer this question of Chinburg in the affirmative.

We consider a specific realization of the renormalization group $(\text{RG})$ transformation acting on functional measures for scalar quantum fields which are expressible as a polymer expansion times an ultra-violet cutoff Gaussian measure. The new and improved definitions and estimates we present are sufficiently general and powerful to allow iteration of the transformation, hence the analysis of complete renormalization group flows, and hence the construction of a variety of scalar quantum field theories.

Abelian closed subgroups of the Galois group of the pythagorean closure of a formally real field are described by means of the inertia group of suitable valuation rings.

We give upper bounds on the modulus of the values at $s\,=\,1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\text{CM}$-fields. For example, we will reduce the determination of all the non-abelian normal $\text{CM}$-fields of degree 24 with Galois group $\text{S}{{\text{L}}_{\text{2}}}\left( {{F}_{3}} \right)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of 23 such $\text{CM}$-fields.

We study the coordinate rings of $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$ scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here ${\mu }'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famous $q$-Kostka polynomial ${{\tilde{K}}_{\lambda \mu }}(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by $\lambda $ in the ring $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$. Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition of ${{\tilde{K}}_{\lambda \mu }}(q)$ as a sum of “atomic” polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model.

Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen [19] imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer [3]. This gives a direct-sum decomposition of the ideals yielding the $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.

Most extant localization theories for spaces, spectra and diagrams of such can be derived from a simple list of axioms which are verified in broad generality. Several new theories are introduced, including localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra, and a theory of localization along a geometric topos morphism. The $f$-localization concept has an analog for simplicial presheaves, and specializes to the ${{\mathbb{A}}^{1}}$-local theory of Morel-Voevodsky. This theory answers a question of Soulé concerning integral homology localizations for diagrams of spaces.

The fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group $U(3)$ in three variables associated with a quadratic extension of $p$-adic fields, and its endoscopic group $U(2)$, by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorphic and admissible representations of $U(3)$ in terms of those of $U(2)$ and base change to $\text{GL(3)}$. It compares the (unstable) orbital integral of the characteristic function of the standard maximal compact subgroup $K$ of $U(3)$ at a regular element (whose centralizer $T$ is a torus), with an analogous (stable) orbital integral on the endoscopic group $U(2)$. The technique is based on computing the sum over the double coset space $T\backslash G/K$ which describes the integral, by means of an intermediate double coset space $H\backslash G/K$ for a subgroup $H$ of $G=U(3)$ containing $T$. Such an argument originates from Weissauer's work on the symplectic group. The lemma is proven for both ramified and unramified regular elements, for which endoscopy occurs (the stable conjugacy class is not a single orbit).

Let $G$ be a free product of cyclic groups of prime order. The structure of the unit group $U(\mathbb{Q}G)$ of the rational group ring $\mathbb{Q}G$ is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of $U(\mathbb{Q}G)$, up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in $\mathbb{Z}G$ is proved. A strong version of the Tits Alternative for $U(\mathbb{Q}G)$ is obtained as a corollary of the structural result.

Generalizing the notion of invariant subspaces on the 2-dimensional torus ${{T}^{2}}$, we study the structure of ${{A}_{\phi }}$-invariant subspaces of ${{L}^{2}}({{T}^{2}})$. A complete description is given of ${{A}_{\phi }}$-invariant subspaces that satisfy conditions similar to those studied by Mandrekar, Nakazi, and Takahashi.

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa (X)\ge 0$, then ${{K}_{X}}L\,\ge \,2q(X)\,-\,4$, where $q\left( X \right)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa (X)=0$ or 1, (2) the case in which $\kappa (X)=2$ and ${{h}^{0}}(L)\,\ge \,2$ , or (3) the case in which $\kappa (X)=2$, $X$ is minimal, ${{h}^{0}}(L)\,=\,1$ , and $L$ satisfies some conditions.

Examples of simple, separable, unital, purely infinite ${{C}^{*}}$-algebras are constructed, including:

1. (1) some that are not approximately divisible;

2. (2) those that arise as crossed products of any of a certain class of ${{C}^{*}}$-algebras by any of a certain class of non–unital endomorphisms;

3. (3) those that arise as reduced free products of pairs of ${{C}^{*}}$-algebras with respect to any from a certain class of states.

We construct the tableaux realization of generalized Verma modules over the Lie algebra $\text{sl(3,}\,\mathbb{C})$. By the same procedure we construct and investigate the structure of a new family of generalized Verma modules over $\text{sl(}n,\,\mathbb{C}\text{)}$.

The purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schrödinger operator in odd dimensions. At the same time we generalize the result to the class of superexponentially decreasing potentials.

Cauchy and Poisson integrals over unbounded sets are employed to prove Mittag-Leffler type theorems with massive singularities as well as approximation theorems for holomorphic and harmonic functions.

We consider the analogue of the $\wedge (p)$―problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the “best possible” result that if $2\,<\,p\,<\,4$ then a random subset of $N$ Legendre polynomials of size ${{N}^{4/p-1}}$ spans an Hilbertian subspace. We also answer a question of König concerning the structure of the space of polynomials of degree $n$ in various weighted ${{L}_{p}}$-spaces.

We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety ${{M}_{\text{nil}}}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

The notion of shape fibration with the near lifting of near multivalued paths property is studied. The relation of these maps—which agree with shape fibrations having totally disconnected fibers—with Hurewicz fibrations with the unique path lifting property is completely settled. Some results concerning homotopy and shape groups are presented for shape fibrations with the near lifting of near multivalued paths property. It is shown that for this class of shape fibrations the existence of liftings of a fine multivalued map is equivalent to an algebraic problem relative to the homotopy, shape or strong shape groups associated.