On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

Motivic torsors

The torsorP _δ=Hom^⊗ (H _DR,H _σ) under the motivic Galois groupG _σ=Aut^⊗ H _δ of the Tannakian category generated by one-motives related by absolute Hodge cycles over a field κ with an embedding σ:k↪ℂ is shown to be determined by its projectionP _σ→P _σ/G _σ ⁰ to a Gal( )-torsor, and by its localizationsP _σ⊕_k k _ξ at a dense subset of orderings ϕ of the field κ, provided κ has virtual cohomological dimension (vcd) one. This result is an application of a recent local-global principle for connected linear algebraic groups over a field κ of vcd ≤1.     

A relative Shafarevich theorem

Suppose given a Galois étale cover YX of proper non-singular curves over an algebraically closed field k of prime characteristic p. Let H be its Galois group, and G any finite extension of H by a p-group P. We give necessary and sufficient conditions on G to be the Galois group of an étale cover of X dominating YX.in final form: 16 September 2003     

Entropy,Periodicity, and Graphs with Zero Euler Characteristic

Let G be a graph which contains exactly one simple closed curve. We prove that a continuous map f : G → G has zero topological entropy if and only if there exist at most k ≤ |(Edg(G) End(G) 3)/2] different odd numbers n1,...,nk such that Per(f) is contained in ∪i=1^k ∪j=0^∞ ni2^j, where Edg(G) is the number of edges of G and End(G) is the number of end points of G.     

Galois groups of intersections of local fields

LetK be a denumerable Hilbertian field with separable algebraic closure and Galois group , letw ₁,...w _n be absolute values on . Then for almost allσ ∈ G _K ⁿ (in the sense of Haar measure) there are no relations between the decomposition groups G _K (ω ₁ σ ₁),...,G _K (w _n σ _n ) of the absolute valuesw ₁ σ ₁,...,w _n σ _n i.e. the subgroup of G _K generated by these groups is the free product of these groups.     

Spectra of ergodic group actions

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ^∞-spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim sup_g→∞Re (g)<1, where is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim sup_g→∞Re (g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.     

On the existence of spines for ${\mathbb{Q}}${\mathbb{Q}}-rank 1 groups

Let X = Γ \G/ K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has -rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D₀. We prove that D₀ is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ -conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ.     

Limit distributions of expanding translates of certain orbits on homogeneous spaces

LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and . LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U⁺ ={g&#x03B5;G:a ⁻ⁿ gaⁿ} →e as n → ∞. Let Ω be a non-empty open subset ofU ⁺ andn _i → ∞ be any sequence. It is showed that . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.     

On a problem of Zambakhidze-Smirnov

We say that the action extension problem is solvable for a bicompact groupG if for any metricG-space and for any topological embeddingc of the orbit spaceX into a metric spaceY there exist aG-space ℤ, an invariant topological embeddingb: → ℤ, and a homeomorphismh: Y → Z such that the diagram is commutative. We prove the following theorem: for a bicompact zero-dimensional groupG, the action extension problem is solvable for the class of dense topological embeddings. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 3–11, July, 1995.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

Approximately isometric lifting in quasidiagonal extensions

Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number ε, two positive elements (projections, partial isometries, unitary elements, respectively) in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of , there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of such that ∥a−b∥ < . As an application, it is shown that for any positive numbers ε and in U(A/I) ₀ , there exists u in U(A)₀ which is a lifting of such that cel(u) < cel. This work was supported by National Natural Science Foundation of China (Grant No. 10771161)     

The Crossed Product by a Partial Endomorphism

Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed) in I , a *-homomorphism α : A → M(I ) and a map L : J → A with some properties, based on earlier works of Pimsner and Katsura, we define a C*-algebra which we call the Crossed Product by a Partial Endomorphism. We introduce the Crossed Product by a Partial Endomorphism induced by a local homeomorphism σ : U → X where X is a compact Hausdorff space and U is an open subset of X. A bijection between the gauge invariant ideals of and the σ, σ^-1- invariant open subsets of X is showed. If (X, σ) has the property that is topologically free for each closed σ, σ^-1-invariant subset X′ of X then we obtain a bijection between the ideals of and the open σ, σ^-1-invariant subsets of X. *Partially supported by CNPq. **Supported by CNPq.     

Homogeneous expansions of normalized biholomorphic convex mappings overB P

The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied. It is proved that the first (k+1) terms of the expansions of the jth componentf j of such a mapf depend only onz _j , for 1 ⩽j⩽n, wherek is the natural number that satisfiesk < ρ ⩽k +I. Whenp→ ∞, this gives the result on the unit polydisc obtained by Suffridge in 1970. Project supported in part by the National Natural Science Foundation of China.     

Measurable orthogonally additive functions modulo a discrete subgroup

Under appropriate conditions on Abelian topological groups G and H, an orthogonality ⊥ ⊂ G ² and a σ-algebra of subsets of G we decompose an -measurable function f: G → H which is orthogonally additive modulo a discrete subgroup K of H into its continuous additive and continuous quadratic part (modulo K). Research supported by the Silesian University Mathematics Department (Functional Equations on Abstract Structures program — the first author, and Iterative Functional Equations and Real Analysis program — the second author).     

Compactness of minimal closed invariant sets of actions of unipotent groups

Let G be a connected Lie group, let Γ be a lattice in G, and let be a unipotent subgroup of G. It is proved that, for the natural action of on G/Γ, every minimal closed -invariant subset is compact. Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

The Boggess-Polking extension theorem for<Emphasis Type="Italic">CR</Emphasis> functions on manifolds with corners

We give the “boundary version” of the Boggess-PolkingCR extension theorem. LetM andN be real generic submanifolds of ℂ ⁿ withN ⊂M and letV be a “wedge” inM with “edge”N and “profile” Σ ⊂T _NM in a neighborhood of a pointz _o.We identify in natural manner and assume that for a holomorphic vector fieldL tangent toM and verifying we have that the Levi form takes a value . Then we prove thatCR functions onV extend ∀_ω to a wedgeV ₁ “attached” toV in direction of a vector fieldiV such that |pr(iV(z ₀))−iv ₀| < ε (where pr is the projection pr:T _NX →T _MX | _N ).We then prove that when the Levi cone “relative to Σ”iZ _Σ = convex hull is open inT _MX, thenCR functions extend to a “full” wedge with edgeN (that is, with a profile which is an open cone ofT _NX). Finally, we prove that iff is defined in a couple of wedges ±V with profiles ±Σ such thatiZ _Σ =T _MX, and is continuous up toN, thenf is in fact holomorphic atz _o.     

Galois embedding problems with cyclic quotient of order<Emphasis Type="Italic">p</Emphasis>

LetK/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable . For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability. Research supported in part by the Natural Sciences and Engineering Research Council of Canada grant R0370A01, as well as by the special Dean of Science Fund at the University of Western Ontario. Supported by the Mathematical Sciences Research Institute, Berkeley. Research supported in part by National Security Agency grant MDA904-02-1-0061.     

The Threshold for the Erdos, Jacobson and Lehel Conjecture to Be True

Let σ（k, n） be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ（k, n） can be realized by a graph containing a clique of k ＋ 1 vertices. Erdos et al. （Graph Theory, 1991, 439-449） conjectured that σ（k, n） = （k - 1）（2n- k） ＋ 2. Li et al. （Science in China, 1998, 510-520） proved that the conjecture is true for k 〉 5 and n ≥ （k2） ＋ 3, and raised the problem of determining the smallest integer N（k） such that the conjecture holds for n ≥ N（k）. They also determined the values of N（k） for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N（k） ≤ （k2） ＋ 3 for k ≥ 8. In this paper, we determine the exact values of σ（k, n） for n ≥ 2k＋3 and k ≥ 6. Therefore, the problem of determining σ（k, n） is completely solved. In addition, we prove as a corollary that N（k） -= [5k-1/2] for k ≥6.