The Russell-Koras contractible threefolds are the smooth affine threefolds having a hyperbolic ${{\mathbb{C}}^{*}}$ -action with quotient isomorphic to the corresponding quotient of the linear action on the tangent space at the unique fixed point. Koras and Russell gave a concrete description of all such threefolds and determined many interesting properties they possess. We use this description and these properties to compute the equivariant Grothendieck groups of these threefolds. In addition, we give certain equivariant invariants of these rings.

We give explicit formulas for the ${{L}_{4}}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials

$$f\left( z \right):=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n}{N} \right){{z}^{n}}.}$$

and

$${{f}_{t}}(z)\,=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n+t}{N} \right){{z}^{n}}.}$$

where $\left( \frac{.}{N} \right)$ is the Jacobi symbol.

Two cases of particular interest are when $N\,=\,pq$ is a product of two primes and $p\,=\,q\,+\,2$ or $p\,=\,q\,+\,4$ . This extends work of Høholdt, Jensen and Jensen and of the authors.

This study arises from a number of conjectures of Erdős, Littlewood and others that concern the norms of polynomials with −1, 1 coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite $N$ .

We use some results about stable relations to show that some of the simple, stable, projectionless crossed products of ${{O}_{2}}$ by $\mathbb{R}$ considered by Kishimoto and Kumjian are inductive limits of type I ${{C}^{*}}$ -algebras. The type I ${{C}^{*}}$ -algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional ${{C}^{*}}$ -algebras.

In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach to show that some given family is topologically trivial. In this approach, we consider a weighted version of $\left( w \right)$ -regularity condition and Kuo’s ratio test condition.

In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra $B$ over $\mathbf{Q}$ ; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on ${{B}^{\times }}$ , analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of ${{B}^{\times }}$ . The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to $\text{CM}$ points on these curves, and are thus isogenous to a product $E\,\times \,E$ , where $E$ is an elliptic curve with complex multiplication. For these $\text{CM}$ points one can make a relation between the action of the $p$ -th Hecke operator and Frobenius at $p$ , similar to the well-known congruence relation of Eichler and Shimura.

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$ , with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$ , we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V\left( \mathbb{A} \right)$ . The Poisson summation formula then yields an identity of distributions on $V\left( \mathbb{A} \right)$ . The truncation used is due to Arthur.

We prove that pre-classifiable (see 3.1) simple nuclear tracially $\text{AF}\,\,{{C}^{*}}$ -algebras $\left( \text{TAF} \right)$ are classified by their $K$ -theory. As a consequence all simple, locally $\text{AH}$ and $\text{TAF}\,\,\,{{C}^{*}}$ -algebras are in fact $\text{AH}$ algebras (it is known that there are locally $\text{AH}$ algebras that are not $\text{AH}$ ). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple $\text{TAF}\,\,\,{{C}^{*}}$ -algebras with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$ -theory and ${{K}_{0}}{{\left( A \right)}_{+}}$ is locally finitely generated, then $A\,\otimes \,Q\,\cong \,B\,\otimes \,Q$ , where $Q$ is the $\text{UHF}$ -algebra with the rational ${{K}_{0}}$ . Classification results (with restriction on ${{K}_{0}}$ - theory) for the above ${{C}^{*}}$ -algebras are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple $\text{TAF}\,\,\,{{C}^{*}}$ -algebras with the unique normalized trace satisfying the $\text{UCT}$ and with ${{K}_{1}}\left( A \right)\,=\,{{K}_{1}}\left( B \right)$ , and $A$ and $B$ have the same rational (scaled ordered) ${{K}_{0}}$ , then $A\,\cong \,B$ . Similar results are also obtained for some cases in which ${{K}_{0}}$ is non-divisible such as ${{K}_{0}}\left( A \right)\,=\,\mathbf{Z}\left[ 1/2 \right]$ .

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\mathbb{C}$ . Let $\text{T}{{\text{r}}_{{{\Lambda }_{1}}\,,\ldots ,\,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}}$ be the characters of the fundamental irreducible representations of $G$ , defined as convergent series on a certain part ${{G}^{\text{tr}-\text{alg}}}\,\subseteq \,G$ . Following Steinberg in the classical case and Brüchert in the affine case, we define the Steinberg map $\chi \,:=\,\left( \text{T}{{\text{r}}_{{{\Lambda }_{1}},\ldots ,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}} \right)$ as well as the Steinberg cross section $C$ , together with a natural parametrisation $\omega :{{\mathbb{C}}^{n}}\times {{\left( {{\mathbb{C}}^{\times }} \right)}^{n-l}}\to C$ . We investigate the local behaviour of $\text{ }\!\!\chi\!\!\text{ }$ on $C$ near $\omega \left( \,\left( 0,\ldots 0 \right)\,\times \,\left( 1,\ldots ,1 \right)\, \right)$ , and we show that there exists a neighborhood of $\left( 0,...,0 \right)\,\,\times \,\,\left( 1,...,1 \right)$ , on which $\text{ }\!\!\chi\!\!\text{ }\,\circ \,\omega $ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on $C$ .

A ${{\mathbb{Z}}_{2}}$ -action with “maximal number of isolated fixed points” (i.e., with only isolated fixed points such that ${{\dim}_{k}}\left( {{\oplus }_{i}}{{H}^{i}}\left( M;k \right) \right)\,\,=\,\,\left| {{M}^{{{\mathbb{Z}}_{2}}}} \right|,\,k\,=\,\left. {{\mathbb{F}}_{2}} \right)$ on a 3-dimensional, closed manifold determines a binary self-dual code of $\text{length}\,\text{=}\,\left| {{M}^{{{\mathbb{Z}}_{2}}}} \right|$ . In turn this code determines the cohomology algebra ${{H}^{*}}\,\left( M;k \right)$ and the equivariant cohomology $H_{{{\mathbb{Z}}_{2}}}^{*}\,\left( M;k \right)$ . Hence, from results on binary self-dual codes one gets information about the cohomology type of 3-manifolds which admit involutions with maximal number of isolated fixed points. In particular, “most” cohomology types of closed 3-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, e.g., one gets that “most” cohomology types (over ${{\mathbb{F}}_{2}}$ ) of closed 3-manifolds do not admit a non-trivial involution.