We study Tychonoff spaces $X$ with the property that, for all topological embeddings $X\,\to \,Y$, the induced map $C(Y)\,\to \,C(X)$ is an epimorphism of rings. Such spaces are called absolute $\mathcal{C}\mathcal{R}$-epic. The simplest examples of absolute $\mathcal{C}\mathcal{R}$-epic spaces are $\sigma $-compact locally compact spaces and Lindelöf $P$-spaces. We show that absolute $\mathcal{C}\mathcal{R}$-epic first countable spaces must be locally compact.

However, a “bad” class of absolute $\mathcal{C}\mathcal{R}$-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute $\mathcal{C}\mathcal{R}$-epic abound, and some are presented.

We prove that it is relatively consistent with $ZFC$ that in any perfect Polish space, for every nonmeager set $A$ there exists a nowhere dense Cantor set $C$ such that $A\,\cap \,C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p\,\le \,x$ for which $\mathbb{Q}\left( {{\pi }_{p}} \right)\,=\,K$, where ${{\pi }_{p}}$ denotes the Frobenius endomorphism of $E$ at $p$. More precisely, under a generalized Riemann hypothesis we show that this number is ${{O}_{E}}\left( {{x}^{17/18}}\,\log x \right)$, and unconditionally we show that this number is ${{O}_{E,K}}\left( \frac{x{{\left( \log \,\log x \right)}^{13/12}}}{{{\left( \log x \right)}^{25/24}}} \right)$ We also prove that the number of imaginary quadratic fields $K$, with − disc $K\,\le \,x$ and of the form $K\,=\,\mathbb{Q}({{\pi }_{p}})$, is ${{\gg }_{E}}\,\log \,\log \,\log \,x$ for $x\,\ge \,{{x}_{0}}\left( E \right)$. These results represent progress towards a 1976 Lang–Trotter conjecture.

Let $k$ be a field of characteristic $0,\,R\,=\,k\left[ {{x}_{1}},\,\ldots ,\,{{x}_{d}} \right]$ be a polynomial ring, and $m$ its maximal homogeneous ideal. Let $I\,\subset \,R$ be a homogeneous ideal in $R$. Let $\lambda (M)$ denote the length of an $R$-module $M$. In this paper, we show that

$$\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( H_{m}^{0}\left( R/{{I}^{n}} \right) \right)}{{{n}^{d}}}\,=\,\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( \text{Ext}_{R}^{d}\left( R/{{I}^{n}},\,R\left( -d \right) \right) \right)}{{{n}^{d}}}$$

always exists. This limit has been shown to be $e(I)/d!$ for $m$-primary ideals $I$ in a local Cohen–Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.

Let $K$ be a function on a unimodular locally compact group $G$, and denote by ${{K}_{n}}\,=\,K\,*\,K\,*\cdots *\,K$ the $n$-th convolution power of $K$. Assuming that $K$ satisfies certain operator estimates in ${{L}^{2}}\left( G \right)$, we give estimates of the norms ${{\left\| {{K}_{n}} \right\|}_{2}}$ and ${{\left\| {{K}_{n}} \right\|}_{\infty }}$ for large $n$. In contrast to previous methods for estimating ${{\left\| {{K}_{n}} \right\|}_{\infty }}$, we do not need to assume that the function $K$ is a probability density or nonnegative. Our results also adapt for continuous time semigroups on $G$. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.

The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer seems to require developments in transcendence theory in characteristic $p$.

Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,\,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Let $L(X)$ be the space of bounded linear operators on the Banach space $X$. We study the strict singularity and cosingularity of the two-sided multiplication operators $S\,\mapsto \,ASB$ on $L(X)$, where $A,\,B\,\in \,L(X)$ are fixed bounded operators and $X$ is a classical Banach space. Let $1\,<\,p\,<\,\infty $ and $p\,\ne \,2$. Our main result establishes that the multiplication $S\,\mapsto \,ASB$ is strictly singular on $L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ if and only if the non-zero operators $A,\,B\,\in \,L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ are strictly singular. We also discuss the case where $X$ is a ${\mathcal{L}^{1}}-$ or a ${{\mathcal{L}}^{\infty }}-$space, as well as several other relevant examples.

We study the semilinear equation

$$-{{\Delta }_{\mathbb{H}}}u(\eta )\,+\,u(\eta )\,=\,f(\eta ,\,\,u(\eta )),\,u\in \,\overset{\circ }{\mathop{S}}\,_{1}^{2}(\Omega ),$$

where $\Omega $ is an unbounded domain of the Heisenberg group ${{\mathbb{H}}^{N}},\,N\,\ge \,1$. The space $\overset{\circ }{\mathop{S}}\,_{1}^{2}(\Omega )$ is the Heisenberg analogue of the Sobolev space $W_{0}^{1,\,2}\,\left( \Omega \right)$. The function $f\,:\,\overset{-}{\mathop{\Omega }}\,\,\times \,\mathbb{R}\,\to \,\mathbb{R}$ is supposed to be odd in $u$, continuous and satisfy some (superlinear but subcritical) growth conditions. The operator ${{\Delta }_{\mathbb{H}}}$ is the subelliptic Laplacian on the Heisenberg group. We give a condition on $\Omega $ which implies the existence of infinitely many solutions of the above equation. In the proof we rewrite the equation as a variational problem, and show that the corresponding functional satisfies the Palais–Smale condition. This might be quite surprising since we deal with domains which are far frombounded. The technique we use rests on a compactness argument and the maximum principle.

We study Dupin hypersurfaces in ${{\mathbb{R}}^{5}}$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.

In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental’ theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$.