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 investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, ${{L}^{1}}\left( G \right)$, and the Fourier algebra, $A\left( G \right)$, of a locally compact group $G$.

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.

Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras ${{\mathfrak{L}}_{q}}$. The center of ${{\mathfrak{L}}_{q}}$ now is generally infinite dimensional.

In this paper, $\mathbb{Z}$-graded Verma modules $\tilde{V}\left( \varphi \right)$ over ${{\mathfrak{L}}_{q}}$ and their corresponding irreducible highest weight modules $V\left( \varphi \right)$ are defined for some linear functions $\varphi $. Necessary and sufficient conditions for $V\left( \varphi \right)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\tilde{V}\left( \varphi \right)$ to be irreducible are obtained.

Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let ${{A}_{\mathbb{R}}}\left( K \right)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior ${{K}^{\circ }}$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $\left( f,\,g \right)$ in ${{A}_{\mathbb{R}}}{{\left( K \right)}^{2}}$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb{C}$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A\left( K \right)$ is 1. Finally, we also characterize all compact real symmetric sets $K$ such that ${{A}_{\mathbb{R}}}\left( K \right)$, respectively ${{C}_{\mathbb{R}}}\left( K \right)$, has Bass stable rank 1.

We give an explicit construction of polynomial (of arbitrary degree) $(\alpha ,\,\beta )$-metrics with scalar flag curvature and determine their scalar flag curvature. These Finsler metrics contain all nontrivial projectively flat $(\alpha ,\,\beta )$-metrics of constant flag curvature.

In the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements ${{H}_{0}}$ and $H$ in the algebra and relate it to the concept of the de la Harpe–Skandalis homotopy invariant determinant associated with piecewise ${{C}^{1}}$-paths of operators joining ${{H}_{0}}$ and $H$. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman–Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier–Fuglede–Kadison differentiation formula.

A set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.

A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

We study $p$-indivisibility of the central values $L\left( 1,\,{{E}_{d}} \right)$ of quadratic twists ${{E}_{d}}$ of a semi-stable elliptic curve $E$ of conductor $N$. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants $d$ splits naturally into several families ${{\mathcal{F}}_{S}}$, indexed by subsets $S$ of the primes dividing $N$. Let ${{\delta }_{S}}={{\gcd }_{d\in {{\mathcal{F}}_{S}}}}L{{(1,{{E}_{d}})}^{\text{alg}}}$, where $L{{(1,{{E}_{d}})}^{\text{alg}}}$ denotes the algebraic part of the central $L$-value, $L(1,\,{{E}_{d}})$. Our main theorem relates the $p$-adic valuations of ${{\delta }_{S}}$ as $S$ varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation between Waldspurger packets on $\widetilde{\text{S}{{\text{L}}_{2}}}$ and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of $p$-indivisibility of $L$-values of quadratic twists.

In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $\text{GU}\left( 1,\,s \right)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.

In the case of $\text{GU}\left( 1,\,2 \right)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

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.

Abstract. Let $H$ be the Hilbert class field of a $\text{CM}$ number field $K$ with maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We evaluate the second term in the Taylor expansion at $s\,=\,0$ of the Galois-equivariant $L$-function ${{\Theta }_{{{S}_{\infty }}\,}}\left( s \right)$ associated to the unramified abelian characters of $\text{Gal}\left( H/K \right)$. This is an identity in the group ring $\mathbb{C}\left[ \text{Gal}\left( H/K \right) \right]$ expressing $\Theta _{{{S}_{\infty }}}^{(n)}\,\left( 0 \right)$ as essentially a linear combination of logarithms of special values $\left\{ \Psi ({{z}_{\sigma }}) \right\}$, where $\Psi :\,{{\mathbb{H}}^{n}}\,\to \,\mathbb{R}$ is a Hilbert modular function for a congruence subgroup of $S{{L}_{2}}\left( {{\mathcal{O}}_{F}} \right)$ and $\left\{ {{z}_{\sigma }}\,:\,\sigma \,\in \,\text{Gal}\left( H/K \right) \right\}$ are $\text{CM}$ points on a universal Hilbert modular variety. We apply this result to express the relative class number ${{h}_{H}}/{{h}_{K}}$ as a rational multiple of the determinant of an $\left( {{h}_{K}}\,-\,1 \right)\,\times \,\left( {{h}_{K}}\,-\,1 \right)$ matrix of logarithms of ratios of special values $\Psi ({{z}_{\sigma }})$, thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for $\Psi ({{z}_{\sigma }})$ in terms of exponentials of special values of $L$-functions.

Let $\mathcal{T}$ be the ${{C}^{*}}$-algebra generated by the Toeplitz operators $\left\{ {{T}_{\varphi }}:\varphi \in {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ on the Hardy space ${{H}^{2}}\left( S \right)$ of the unit sphere in ${{C}^{n}}$. It is well known that $\mathcal{T}$ is contained in the essential commutant of $\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$. We show that the essential commutant of $\left\{ {{T}_{\varphi }}:\varphi \in \text{VMO}\cap {{L}^{\infty }}\left( S,d\sigma \right) \right\}$ is strictly larger than $\mathcal{T}$.

In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series. To do that we need to normalize using the Langlands–Shahidi's normalization arising from the twisted endoscopy and the associated representations of the general linear group.

In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.

Let ${{G}_{n}}=\text{S}{{\text{p}}_{n}}\left( F \right)$ be the rank $n$ symplectic group with entries in a nondyadic $p$-adic field $F$. We further let ${{\tilde{G}}_{n}}$ be the metaplectic extension of ${{G}_{n}}\,\text{by}\,{{\mathbb{C}}^{1}}=\left\{ z\in {{\mathbb{C}}^{\times }}|\,|z|=1 \right\}$ defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of ${{\tilde{G}}_{2}}$. In most cases, we will use some techniques developed by Tadić that analyze the Jacquet modules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters $\chi $ with ${{\chi }^{2}}=1$. Because such representations $\pi $ are unitary, to show the irreducibility of $\pi $, it suffices to show that ${{\dim}_{\mathbb{C}}}\,\text{Ho}{{\text{m}}_{{\tilde{G}}}}\left( \pi ,\,\pi \right)\,=\,1$ . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of ${{\tilde{G}}_{2}}.$ We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in $\text{Ho}{{\text{m}}_{{{{\tilde{G}}}_{2}}}}\left( \pi ,\,\pi \right)$ for the $\pi $ in question.

We calculate the group of homotopy classes of homotopy self-equivalences of 4-manifolds with free fundamental group and obtain a classification of such 4-manifolds up to s-cobordism.

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 $X$.

A new decomposition, the mutually aposyndetic decomposition of homogeneous continua into closed, homogeneous sets is introduced. This decomposition is respected by homeomorphisms and topologically unique. Its quotient is a mutually aposyndetic homogeneous continuum, and in all known examples, as well as in some general cases, the members of the decomposition are semi-indecomposable continua. As applications, we show that hereditarily decomposable homogeneous continua and path connected homogeneous continua are mutually aposyndetic. A class of new examples of homogeneous continua is defined. The mutually aposyndetic decomposition of each of these continua is non-trivial and different from Jones’ aposyndetic decomposition.

In this paper we establish a priori${{h}^{1}}$-estimates in a bounded domain for parabolic equations with vanishing $\text{LMO}$ coefficients.