On the G-compactifications of the rational numbers

We show that for any sufficiently homogeneous metrizable compactum X there is a Polish group G acting continuously on the space of rational numbers \mathbbQ{\mathbb{Q}} such that X is its unique G-compactification. This allows us to answer Problem 995 in the ‘Open Problems in Topology II’ book in the negative: there is a one-dimensional Polish group G acting transitively on \mathbbQ{\mathbb{Q}} for which the Hilbert cube is its unique G-completion.     

Groups whose non-linear irreducible characters are rational valued

A finite group G all of whose nonlinear irreducible characters are rational is called a \mathbbQ₁{\mathbb{Q}_1}-group. In this paper, we obtain some results concerning the structure of \mathbbQ₁{\mathbb{Q}_1}-groups.     

Reductions of points on elliptic curves

Let E be an elliptic curve defined over \mathbbQ{\mathbb{Q}}. Let Γ be a subgroup of rank r of the group of rational points E(\mathbbQ){E(\mathbb{Q})} of E. For any prime p of good reduction, let [`(G)]{\bar{\Gamma}} be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have

Fields of moduli and definition of hyperelliptic curves of odd genus

Mestre has shown that if a hyperelliptic curve C of even genus is defined over a subfield k ì \mathbbC{k \subset \mathbb{C}} then C can be hyperelliptically defined over the same field k. In this paper, for all genera g > 1, g o 1{g\equiv1} mod 4, hence odd, we construct an explicit hyperelliptic curve defined over \mathbbQ{\mathbb{Q}} which can not be hyperelliptically defined over \mathbbQ{\mathbb{Q}}.     

Characterizing maximal compact subgroups

We prove that for a compact subgroup H of a locally compact almost connected group G, the following properties are mutually equivalent: (1) H is a maximal compact subgroup of G, (2) the coset space G/H is \mathbbQ{\mathbb{Q}} -acyclic and \mathbbZ/2\mathbbZ{\mathbb{Z}/2\mathbb{Z}} -acyclic in Čech cohomology, (3) G/H is contractible, (4) G/H is homeomorphic to a Euclidean space, (5) G/H is an absolute extensor for paracompact spaces, (6) G/H is a G-equivariant absolute extensor for paracompact proper G-spaces having a paracompact orbit space.     

Stable systolic category of manifolds and the cup-length

It follows from a theorem of Gromov that the stable systolic category cat_stsys M{\rm cat}_{\rm stsys} M of a closed manifold M is bounded from below by cl_\mathbbQ M{\rm cl}_{\mathbb{Q}} M, the rational cup-length of M [Ka07]. We study the inequality in the opposite direction. In particular, combining our results with Gromov’s theorem, we prove the equality cat_stsys M = cl_\mathbbQ M{\rm cat}_{\rm stsys} M = {\rm cl}_{\mathbb{Q}} M for simply connected manifolds of dimension ≤ 7.     

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbF_m,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbF_m,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbF_m,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbF_m,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?_j=1^m X_j²{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d ^2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X ₁, . . . , X _m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

Minimal subsystems of affine dynamics on local fields

We describe the dynamics of an arbitrary affine dynamical system on a local field by exhibiting all its minimal subsystems. In the special case of the field \mathbbQ_p{\mathbb{Q}_p} of p-adic numbers, for any non-trivial affine dynamical system, we prove that the field \mathbbQ_p{\mathbb{Q}_p} is decomposed into a countable number of invariant balls or spheres each of which consists of a finite number of minimal subsets. Consequently, we give a complete classification of topological conjugacy for non-trivial affine dynamics on \mathbbQ_p{\mathbb{Q}_p} . For each given prime p, there is a finite number of conjugacy classes.     

Local–global divisibility by 4 in elliptic curves defined over {\mathbb{Q}}

Let ${\mathcal{E}}Let E{\mathcal{E}} be an elliptic curve defined over \mathbbQ{\mathbb{Q}} . Let P ? E(\mathbb Q){P\in {\mathcal{E}}(\mathbb {Q})} and let q be a positive integer. Assume that for almost all valuations v ? \mathbbQ{v\in \mathbb{Q}} , there exist points D_v ? E(\mathbb Q_v){D_v\in {\mathcal{E}}(\mathbb {Q}_v)} such that P = qD _v . Is it possible to conclude that there exists a point D ? E(\mathbb Q){D\in {\mathcal{E}}(\mathbb {Q})} such that P = qD? A full answer to this question is known when q is a power of almost all primes p ? \mathbbN{p\in \mathbb{N}} , but some cases remain open when p ? S={2,3,5,7,11,13,17,19,37,43,67,163}{p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}} . We now give a complete answer in the case when q = 4.     

On the distribution of powers of a complex number

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo \mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and \mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo \mathbbZ[i]{\mathbb{Z}[i]} the fractional part of z and write {z} for this, in general, complex number lying in the unit square S:={z ? \mathbbC:0 ￡ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over \mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ α ⁿ  +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤  1 and x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence z_n ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ α ⁿ −z _n , n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at (1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large.     

Algebraic characterization of the isometries of the hyperbolic 5-space

Let \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} be the group of invertible 2 × 2 matrices over the division algebra \mathbbH{\mathbb{H}} of quaternions. \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} .     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Metrizability of the space of {\mathbb{R}} -places of a real function field

For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x₁,?, x_n){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.     

Sharply k-Transitive Permutation Groups Viewed as Galois Groups

The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), and Zassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers \mathbbQ{\mathbb{Q}}. In this article, we study some properties of the polynomials f ? \mathbbQ[x]{f \in \mathbb{Q}[x]} such that the Galois group Gal(f) acts sharply k-transitively on its roots.     

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

A New Class of Inner Functions Uniformly Approximable by Interpolating Blaschke Products

We study the class of inner functions Q{\Theta} whose zero set Z(Q){Z(\Theta)} stays hyperbolically close to [`(Z_\mathbbD(Q))]{\overline{Z_\mathbb{D}(\Theta)}} on the corona of H ^∞ and show that these functions are uniformly approximable by interpolating Blaschke products.     

All 4-Edge-Connected HHD-Free Graphs are {\mathbb{Z}_3}-Connected

An undirected graph G = (V, E) is called \mathbbZ₃{\mathbb{Z}_3}-connected if for all b: V ? \mathbbZ₃{b: V \rightarrow \mathbb{Z}_3} with ?_{v ? V}b(v)=0{\sum_{v \in V}b(v)=0}, an orientation D = (V, A) of G has a \mathbbZ₃{\mathbb{Z}_3}-valued nowhere-zero flow f: A? \mathbbZ₃-{0}{f: A\rightarrow \mathbb{Z}_3-\{0\}} such that ?_{e ? d⁺(v)}f(e)-?_{e ? d^-(v)}f(e)=b(v){\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)} for all v ? V{v \in V}. We show that all 4-edge-connected HHD-free graphs are \mathbbZ₃{\mathbb{Z}_3}-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the \mathbbZ₃{\mathbb{Z}_3}-connectivity for 4-edge-connected chordal graphs.     

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of \mathbbG_\texta² \mathbb{G}_{\text{a}}^2 , to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \mathbbG_\texta {\mathbb{G}_{\text{a}}} ⋊ \mathbbG_\textm {\mathbb{G}_{\text{m}}} . Bibliography: 32 titles.