For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of ${{\mathbb{R}}^{n}}$ is closed, connected, and locally convex, then it is convex. We give an analogous “local to global convexity” theorem when the inclusion map of $X$ to ${{\mathbb{R}}^{n}}$ is replaced by a map from a topological space $X$ to ${{\mathbb{R}}^{n}}$ that satisfies certain local properties. Our motivation comes from the Condevaux–Dazord–Molino proof of the Atiyah–Guillemin–Sternberg convexity theorem in symplectic geometry.

A triangulation of a hyperbolic 3-manifold is $L$ -thick if each tetrahedron having all vertices in the thick part of $M$ is $L$ -bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic 3-manifold has an $L$ -thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of ${{\pi }_{1}}$ -injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds.

The fractional Galois ideal is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher $K$ -groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.

Correspondence between torsion-free connections with nilpotent skew-symmetric curvature operator and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

S. Stahl conjectured that the zeros of genus polynomials are real. In this note, we disprove this conjecture.

Let $E$ be an elliptic curve, and let ${{L}_{n}}$ be the Kummer extension generated by a primitive ${{p}^{n}}$ -th root of unity and a ${{p}^{n}}$ -th root of $a$ for a fixed $a\,\in \,{{\mathbb{Q}}^{\times }}\,-\,\left\{ \pm 1 \right\}$ . A detailed case study by Coates, Fukaya, Kato and Sujatha and $V$ . Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over ${{L}_{n}}$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations.

We address the problem of computing the fundamental group of a symplectic ${{S}^{1}}$ -manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian ${{S}^{1}}$ -action. Several examples are presented to illustrate our main results.

Recently, Granville and Soundararajan have made fundamental breakthroughs in the study of character sums. Building on their work and using estimates on short character sums developed by Graham–Ringrose and Iwaniec, we improve the Pólya–Vinogradov inequality for characters with smooth conductor.

Let $\mathbb{G}$ be a step-two nilpotent group of $\text{H}$ -type with Lie algebra $\mathfrak{G}\,=\,V\,\oplus \,\text{t}$ . We define a class of vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ on $\mathbb{G}$ depending on a real parameter $k\,\ge \,1$ , and we consider the corresponding $p$ -Laplacian operator ${{L}_{p,\,k}}u\,=\,di{{v}_{X}}\left( {{\left| {{\nabla }_{X}}u \right|}^{p-2}}{{\nabla }_{X}}u \right)$ . For $k\,=\,1$ the vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$ ; for $\mathbb{G}$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator ${{L}_{p,\,k}}$ and as an application, we get a Hardy type inequality associated with $X$ .

Let $\mathcal{F}$ be a coherent rank 2 sheaf on a scheme $Y\,\subset \,{{\mathbb{P}}^{n}}$ of dimension at least two and let $X\,\subset \,Y$ be the zero set of a section $\sigma \,\in \,{{H}^{0}}\left( \mathcal{F} \right)$ . In this paper, we study the relationship between the functor that deforms the pair $\left( \mathcal{F},\,\sigma\right)$ and the two functors that deform $\mathcal{F}$ on $Y$ , and $X$ in $Y$ , respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of e.g., the moduli scheme ${{\text{M}}_{\text{Y}}}\left( P \right)$ of stable sheaves on a threefold $Y$ at $\left( \mathcal{F} \right)$ , and the scheme structure at ( $X$ ) of the Hilbert scheme of curves on $Y$ become closely related. Using this relationship, we get criteria for the dimension and smoothness of ${{\text{M}}_{\text{Y}}}\left( P \right)$ at $\left( \mathcal{F} \right)$ , without assuming $\text{Ex}{{\text{t}}^{2}}\left( \mathcal{F},\,\mathcal{F} \right)\,=\,0$ . For reflexive sheaves on $Y\,=\,{{\mathbb{P}}^{3}}$ whose deficiency module $M\,=\,H_{*}^{1}\left( \mathcal{F} \right)$ satisfies $_{0}\text{Ex}{{\text{t}}^{2}}\left( M,\,M \right)\,=\,0$ (e.g., of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of $H_{*}^{0}\left( \mathcal{F} \right)$ . Moreover, we show that every irreducible component of ${{\text{M}}_{\mathbb{P}}}^{3}\left( P \right)$ containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of ${{\text{M}}_{\mathbb{P}}}^{3}\left( P \right)$ that contains a reflexive stable sheaf with “small” deficiency module $M$ .

We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of $L$ -functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.

Furthermore, as arithmetical applications, we make a conjecture on the distribution of ${{a}_{p}}$ 's amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).

A borderline case function $f$ for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta \,>\,0$ . On its support ${{I}_{0}}\,=\,{{\left[ 0,\,1 \right]}^{n}},\,f\left( x \right)$ can be expressed by the binary expansions of the coordinates of $x$ . In particular, $f\,=\,{{f}_{\beta }}\,\in \,{{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ if and only if $\alpha \,<\,\beta \,<\frac{n}{2}$ , while for $\beta \,=\,\alpha $ , it was shown by Yue and Dafni that $f$ satisfies a John–Nirenberg inequality for ${{Q}_{\alpha }}\left( {{\mathbb{R}}^{n}} \right)$ . When $\beta \,\ne \,1$ , $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside ${{I}_{0}}$ . In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta \,\in \,\left( 0,\,1 \right)$ , $f$ is onto from ${{I}_{0}}$ to $\left[ -\frac{1}{1-{{2}^{-\beta }}},\,\frac{1}{1-{{2}^{-\beta }}} \right]$ , and the graph of $f$ has a non-integer fractal dimension $n\,+\,1\,-\beta$ .