Weak Covering Properties of Weak Topologies

We consider covering properties of weak topologies of Banachspaces, especially of weak or point-wise topologies of functionspaces C(K), for compact spaces K. We answer questions posedby A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Ourmain results are the following. A Banach space of density atmost ₁ is hereditarily metaLindel of in its weak topology. Ifthe weight of a compact spaceK is at most ₁, then the spacesC_w(K) and C_p(K) are hereditarily metaLindel of. Let be the one-point compactificationof a treeT. Then the space is hereditarily -metacompact. If T is an infinitely branchingfull tree of uncountable height and of cardinality bigger thanc, then the weak topology of the unit sphere of is not -fragmented by any metric. The space C_p(rß₁)is neither metaLindel of nor -relatively metacompact. The spaceC_p(rß₂) is not -relatively metaLindel of. Under theset-theoretic axiom , there exists a scattered compact spaceK₁ such that the space C_p(K₁) is not -relatively metacompact,and under a related axiom , there exists a scattere compactspace K₂ such that the space C_p(K₂) is not -relatively metaLindelof. 1991 Mathematics Subject Classification: 54C35, 46B20, 54E20,54D30.     

Wild Recurrent Critical Points

It is conjectured that a rational map whose coefficients arealgebraic over Q_p has no wandering components of the Fatou set.Benedetto has shown that any counterexample to this conjecturemust have a wild recurrent critical point. We provide the firstexamples of rational maps whose coefficients are algebraic over Q_p and that have a (wild) recurrent critical point. In fact,it is shown that there is such a rational map in every one-parameterfamily of rational maps that is defined over a finite extensionof Q_p and that has a Misiurewicz bifurcation.     

Convex Bodies with Homothetic Sections

We prove that if K is a convex body in Eⁿ⁺¹, n2, and p₀ is apoint of K with the property that all n-sections of K throughp₀ are homothetic, then K is a Euclidean ball.     

The Minimum Index of a Non-Congruence Subgroup of SL2 Over an Arithmetic Domain. II: The Rank Zero Cases

Let K be a function field of genus g with a finite constantfield F_q. Choose a place of K of degree and let C be the arithmeticDedekind domain consisting of all elements of K that are integraloutside . An explicit formula is given (in terms of q, g and) for the minimum index of a non-congruence subgroup in SL₂(C).It turns out that this index is always equal to the minimumindex of an arbitrary proper subgroup in SL₂(C). The minimumindex of a normal non-congruence subgroup is also determined.     

Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

We show that for every natural number m a finitely generatedmetabelian group G embeds in a quotient of a metabelian groupof type FP_m. Furthermore, if m 4, the group G can be embeddedin a metabelian group of type FP_m. For L a finitely generatedmetabelian Lie algebra over a field K and a natural number mwe show that, provided the characteristic p of K is 0 or p >m, then L can be embedded in a metabelian Lie algebra of typeFP_m. This result is the best possible as for 0 < p m everymetabelian Lie algebra over K of type FP_m is finite dimensionalas a vector space.     

A valuation criterion for normal bases in elementary abelian extensions

Let p be a prime number, and let K be a finite extension ofthe field _p of p-adic numbers. Let N be a fully ramified, elementaryabelian extension of K. Under a mild hypothesis on the extensionN/K, we show that every element of N with valuation congruentmod [N:K] to the largest lower ramification number of N/K generatesa normal basis for N over K.     

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.     

A Uniqueness Problem in Valued function Fields of Conics

Let v₀ be a valuation of a field K₀ with value group G₀. LetK be a function field of a conic over K₀, and let v be an extensionof v₀ to K with value group G such that G/G₀ is not a torsiongroup. Suppose that either (K₀, v₀) is henselian or v₀ is ofrank 1, the algebraic closure of K₀ in K is a purely inseparableextension of K₀, and G₀ is a cofinal subset of G. In this paper,it is proved that there exists an explicitly constructible elementt in K, with v(t) non-torsion modulo G₀ such that the valuationof K₀(t), obtained by restricting v, has a unique extensionto K. This generalizes the result proved by Khanduja in theparticular case, when K is a simple transcendental extensionof K₀ (compare [4]). The above result is an analogue of a resultof Polzin proved for residually transcendental extensions [8].     

ON THE INTEGRAL GALOIS MODULE STRUCTURE OF CYCLIC EXTENSIONS OF p-ADIC FIELDS

We strengthen results of Miyata on the integral Galois modulestructure of totally ramified cyclic Kummer extensions K ofdegree pⁿ of a p-adic field k. Let c₁(K/k) be the first ramificationnumber of K/k, and let c(K/k) be the least non-negative residueof c₁(K/k) modulo pⁿ. Suppose that K is of the form k() with^pn k and val _K(–1)>0, (val _K(–1), p)= 1. Thenthe valuation ring of K is free over its associated order ifc(K/k) divides p^m–1 for some m with 1mn; the converseholds if n= 2; and is a Hopf order (or a Gorenstein order)if and only if c(K/k) = pⁿ–1.     

The Manin-Mumford Conjecture: A Brief Survey

The Manin–Mumford conjecture asserts that if K is a fieldof characteristic zero, C a smooth proper geometrically irreduciblecurve over K, and J the Jacobian of C, then for any embeddingof C in J, the set C(K)J(K)_tors is finite. Although the conjecturewas proved by Raynaud in 1983, and several other proofs haveappeared since, a number of natural questions remain open, notablyconcerning bounds on the size of the intersection and the completedetermination of C(K)J(K)_tors for special families of curvesC. The first half of this survey paper presents the Manin–Mumfordconjecture and related general results, while the second describesrecent work mostly dealing with the above questions. 1991 MathematicsSubject Classification 11G10, 11G30, 11G35, 14G25, 14H25, 14H40.     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

Definable groups and compact p-adic Lie groups

We formulate p-adic analogues of the o-minimal group conjecturesfrom the works of Hrushovski, Peterzil and Pillay [J. Amer.Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162];that is, we formulate versions that are appropriate for groupsG definable in (saturated) P-minimal fields. We then restrictour attention to saturated models K of Th(_p) and Th(_{p, an}),record some elementary observations when G is defined over thestandard model _p, and then make a detailed analysis of the casewhere G = E(K) for E an elliptic curve over K. Essentially,our P-minimal conjectures hold in these contexts and, moreover,our case study of elliptic curves yields counterexamples toa more naive direct translation of the o-minimal conjectures.     

INVARIANT FIELDS AND LOCALIZED INVARIANT RINGS OF p-GROUPS

It is well known that for a p-group, the invariant field ispurely transcendental (T. Miyata, Invariants of certain groupsI, Nagoya Math. J. 41 (1971), 69–73). In this note, weshow that a minimal generating set of this field can be chosenas homogeneous invariants from the invariant ring. As a result,we show that the invariant ring localized at one suitable invariantis the localization of a polynomial subring at this same invariant.This second result is a generalization of a recent result ofthe first author for cyclic groups of order p (H. E. A. Campbell,Rings of invariants of representations of C_p in characteristicp, preprint, 2006). As well, we specialize these results tothis latter case.     

K0 and the dimension filtration for p-torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoreticquestions related to the representation theory of the completedgroup algebra kG of G with coefficients in a finite field kof characteristic p. We show that if M is a finitely generatedkG-module with canonical dimension smaller than the dimensionof the centralizer, as a p-adic analytic group, of any p-regularelement of G, then the Euler characteristic of M is trivial.Writing _i for the abelian category consisting of all finitelygenerated kG-modules of dimension at most i, we provide an upperbound for the rank of the natural map from the Grothendieckgroup of _i to that of _d, where d denotes the dimension of G.We show that this upper bound is attained in some special cases,but is not attained in general.     

On a Theorem of Childs on Normal Bases of Rings of Integers: Addendum

Let p 3 be a prime number, F be a number field with _p F^x,and K = F(_p). In a previous paper, the author proved, undersome assumption on p and F, that an unramified cyclic extensionN/F of degree p has a normal integral basis if and only if thepushed-up extension NK/K has a normal integral basis. This addendumshows that the assertion holds without the above-mentioned assumption.     

Singularities and Limit Functions in Iteration of Meromorphic Functions

Let f(z) be a transcendental meromorphic function. The paperinvestigates, using the hyperbolic metric, the relation betweenthe forward orbit P(f) of singularities of f^–1 and limitfunctions of iterations of f in its Fatou components. It ismainly proved, among other things, that for a wandering domainU, all the limit functions of {fⁿ|_U} lie in the derived setof P(f) and that if f^np|_V q(n +) for a Fatou component V, theneither q is in the derived set of S_p (f) or f^p(q) = q. As applicationsof main theorems, some sufficient conditions of the non-existenceof wandering domains and Baker domains are given.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

From Trace States to States on the K0 Group of a Simple C*-Algebra

It is shown that any continuous affine surjection from a metrizableChoquet simplex onto a compact convex set occurs as the restrictionmap from the tracial state space onto the state space of theK₀ group of a separable unital simple C^*-algebra which is theinductive limit of a sequence of subhomogeneous C^*-algebras     

A Characterization of Finite Tame Extensions

Let v be a henselian valuation of a field K. In this paper itis proved that any finite extension (K', v') of (K, v) is tameif and only if there exists 0 in K' such that v'() = v(Tr_K'/K())using elementary results of valuation theory. A special caseof this result, when the characteristic of the residue fieldof v is p > 0 and (K', v')/(K, v) is an extension of degreep, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math.404 (1990) 1–38). 1991 Mathematics Subject Classification12J10, 12J25, 13A18.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.