Pour toute sous-vari ét é géométriquement irréductible $V$ du groupe multiplicatif $\mathbb{G}_{m}^{n}$, on sait qu’en dehors d’un nombre fini de translat és de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minorée par une certaine quantité $q{{(V)}^{-1}}>0$. On connaît de plus une borne supérieure pour la somme des degrés de ces translatés de tores pour des valeurs de $q(V)$ polynomiales en le degré de $V$. Ceci n’est pas le cas si l’on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement linéaire en l’inverse du degré.

Nous apportons ici une réponse partielle à ce problème : nous donnons une majoration de la somme des degrés de ces translat és de sous-tores de codimension 1 d’une hypersurface $V$. Les résultats, obtenus dans le cas de $\mathbb{G}_{m}^{3}$,mais complètement explicites, peuvent toutefois s’étendre à $\mathbb{G}_{m}^{n}$,moyennant quelques petites complications inhérentes à la dimension $n$.

In this paper we study the extension problem, the averaging problem, and the generalized Erdős–Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number $\ell $ and suppose $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ for a set of primes ${\mathfrak {p}} \subset {\mathcal {O}_{K}}$ of density 1. When $\ell =2$ Serre has shown that there does not necessarily exist a K-isogenous $A'$ such that $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$ . We extend those results to all odd $\ell $ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod- $\ell ^2$ representation.

We study systems of two-tangle equations

$$ \begin{align*}\begin{cases} N(X+T_1)=L_1,\\ N(X+T_2)=L_2, \end{cases}\end{align*} $$

We show that every system of framed tangle equations has at most one-framed rational solution. Furthermore, we show that the Jones unknot conjecture implies that if a system of tangle equations has a rational solution, then that solution is unique among all two-tangles. This result potentially opens a door to a purely topological disproof of the Jones unknot conjecture.

We introduce the notion of the Kauffman bracket ratio $\{T\}_q\in \mathbb Q(q)$ of any two-tangle T and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic T. We also prove that for rational T, the brackets $\{T\}_q$ coincide with the q-rationals of Morier-Genoud and Ovsienko.

Additionally, we relate systems of tangle equations to the cosmetic surgery conjecture and the nugatory crossing conjecture.

In recent years the harmonic analysis of n-fold (n > 2) metaplectic coverings of GL2 has played an increasingly important role in certain aspects of algebraic number theory. In large part this has been inspired by the pioneering work of Kubota (see [3] for example); as an application one could cite the solution by Heath-Brown and Patterson [3] to a question of Kummer's on the distribution of the arguments of cubic Gauss sums. In that paper, Eisenstein series on the 3-fold metaplectic cover of GL2(A) play a crucial role.

The object of this note is to point out that the theory of Eisenstein series can be made to work for a wide class of finite central coverings. Indeed, once the assumptions are made, the usual theory carries over readily, and one obtains a spectral decomposition of the appropriate L2-space of functions; this is done in Section 2 of this paper.

In a recent paper Mostert and Shields (4) showed that if a space homeomorphic to the non-negative real numbers is a certain type of topological semigroup, then the semigroup must be that of the non-negative real numbers with the usual multiplication. Somewhat earlier Faucett (2) showed that if a compact connected ordered space is a suitably restricted topological semigroup, then it must be both topologically and algebraically the same as the unit interval of real numbers with its usual multiplication.

In studying certain binary relations on topological spaces there have become known (see, in particular, Wallace (5) and the author (3)) a number of properties analogous to those possessed by topological semigroups. Because of these analogous properties between relations and semigroups the author was motivated by the general nature of the Faucett and Mostert-Shields results (that is, that the multiplication assumed turned out to be the same as the usual multiplication) to feel that certain relations on a connected ordered space should turn out to be the same as the orders whose order topologies are the topology on the space.

We explicitly find regulators of an infinite family $\{{{L}_{m}}\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring ${{\mathbb{F}}_{q}}\left( t \right)$, where ${{\mathbb{F}}_{q}}$ is the finite field of order $q$ with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having the same conductors. We obtain a lower bound on the class numbers of the family $\{{{L}_{m}}\}$ and some result on the divisibility of the divisor class numbers of cyclotomic function fields that contain $\{{{L}_{m}}\}$ as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field ${{\mathbb{F}}_{q}}\left( t \right)$ in $\{{{L}_{m}}\}$.

Let Bn denote the unit ball in Cn with boundary S. We will be concerned with spaces of holomorphic functions on Bn and will use much of the notation and terminology found in W. Rudin's book [16]. Thus, if f is holomorphic in Bn and has homogeneous polynomial expansion

the radial derivative of f is given by

We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is controlled by a pair of invariants, called the generalized functional and generalized homological invariants, and we derive an explicit birational model for them, which we call the Weierstrass form. We then describe how to resolve the singularities of the Weierstrass form to obtain the “minimal form”, which has mild singularities and is unique up to birational maps in codimension 2. Finally, we describe some of the geometric properties of threefolds in minimal form, including their singular fibres, canonical divisor, and Betti numbers.

Let I be an interval, and let for 1 ≦ j ≦ I < ∞ be abutted subintervals such that . Let τj be a linear differential expression defined on Ij. In this paper we study densely defined operators associated with

(0.1)

Let $\mathcal{A}$ be a Banach algebra with a bounded right approximate identity and let $\mathcal{B}$ be a closed ideal of $\mathcal{A}$. We study the relationship between the right identities of the double duals ${{\mathcal{B}}^{*}}^{*}$ and ${{\mathcal{A}}^{**}}$ under the Arens product. We show that every right identity of ${{\mathcal{B}}^{*}}^{*}$ can be extended to a right identity of ${{\mathcal{A}}^{**}}$ in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra $A\left( G \right)$ of a locally compact group $G$, an element $\phi \in A{{\left( G \right)}^{**}}$ is in $A\left( G \right)$ if and only if $A\left( G \right)\phi \subseteq A\left( G \right)$ and $E\phi =\phi$ for all right identities $E$ of $A{{\left( G \right)}^{**}}$. We also prove some results about the topological centers of ${{\mathcal{B}}^{**}}$ and ${{\mathcal{A}}^{**}}$.

Let A be a C*-algebra and H a Hilbert space of large enough (infinite at least) dimension so that every πƒ, where ƒ is a factor state on A, can be unitarily represented on H. Let Fac (A, H) denote the set of all factor representations of A on H. If π is in Fac (A, H) we call its essential subspace the smallest, closed, vector subspace KoiH such that π (A ) is null on H Θ K. We define Fac∞(A, H) to be the set of elements in Fac (A, H) whose essential subspace is H.

Let KG be the group ring of a group G over a field K, and U its group of units. Given a group H, we shall denote by ξ(H) the center of H and by T(H) the set of all its torsion elements.

The following question appears in [5, p. 231]: When is Un ⊂ ξ (U), for some n? It was considered by G. Cliff and S. K. Sehgal in [1], where G is assumed to be a solvable group. A complete answer at characteristic zero is given there. Also they obtain partial results at characteristic p ≠ 0, with certain restrictions on the exponent n.

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold, Mn, in Euclidean space, which are sufficient to prove that Mn has a transverse field (see 1.1 and 1.2, definitions).

In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifold Mn.

Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).

After the research for this paper was completed, a paper of Davis (2) came to our attention.

1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank)(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B-l) and that B ⇒ A if and only if F = AB-1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an† is summable A if its sequence of partial sums is A-limitable.

We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$, where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

We investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

Let Ω be a cyclic extension of degree l over the field Σ. Correcting an error which for some time had been haunting the literature, Hasse (1, p. 272) noted that for l = 2, the field Ω may contain a unit ξ such that

.

We prove that a Hamiltonian $p\in {{C}^{\infty }}({{T}^{*}}{{\mathbf{R}}^{n}})$ is locally integrable near a non-degenerate critical point ${{\rho }_{0}}$ of the energy, provided that the fundamental matrix at ${{\rho }_{0}}$ has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in the ${{C}^{\infty }}$ sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that when $p$ is holomorphic near ${{\rho }_{0}}\in {{T}^{*}}{{\mathbf{C}}^{n}},$ then Re $p$ becomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge, i.e.,$p$ may not be integrable. These normal forms also hold in the semi-classical frame.

In this paper we continue our investigation of the class of right LCM domains which was introduced in [2]. A right LCM domain is an (not necessarily commutative) integral domain with unity in which the intersection of any two principal right ideals is again principal. In this note we study the right quotient rings of such a ring. In Section 1 we describe some of the characteristic properties of right quotient monoids with respect to which quotient rings are formed. Three particular types of quotient rings are described in Section 2. In Section 3 we relate the right ideals of a ring to those of its quotient ring.