Realisierung endlicher Gruppen als Galoisgruppen

A talk at the 1984 Oberwolfach meeting on Algebraic Number Theory will be summarized. It surveyed some new results on the realization of finite groups as Galois groups over the fields and ^ab, where ^ab is the maximal abelian extension field of .     

Solving systems of polynomial inequalities over a real closed field in subexponential time

Let _m= (₁,..., _m) denote an ordered field, where _i+1>0 is infinitesimal relative to the elements of _i, 0 < –i < m (by definition, ₀= ). Given a system of inequalities f₁ > 0, ..., f_s > 0, f_s+1 0, ..., f_k 0, where f_{j m} [X₁,..., X_n] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the f_js is at most 2^M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of _m in polynomial time with respect to M, ((d)ⁿd₀)^n+m. In the case _m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d ₀ ^{m 2U(n)} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988.     

Unique <Emphasis Type="Italic">a</Emphasis>-closure for some ℓ-groups of rational valued functions

Usually, an abelian -group, even an archimedean -group, has a relatively large infinity of distinct a-closures. Here, we find a reasonably large class with unique and perfectly describable a-closure, the class of archimedean -groups with weak unit which are -convex. ( is the group of rationals.) Any C(X, ) is -convex and its unique a-closure is the Alexandroff algebra of functions on X defined from the clopen sets; this is sometimes C(X).     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

Canonical Equivalence Relations on ℚ n

We show that for each positive integer n there is a finite list of equivalence relations on [] ⁿ with the property that for every other equivalence relation E on [] ⁿ there is X of order type equal to the order type of , such that E[X] ⁿ is equal to one of the equivalence relations from the list.     

A topological characterization for closed sets under polar duality in ℚ n

A topological characterization is given for closed sets in ⁿ under the restriction of (cone) polar duality to ⁿ .     

Über lineare Unabhängigkeit

It is the aim of the present work to prove, under appropriate conditions, lower estimates for the dimension of w ₁ + ... + w _m over , wherew ₁,...,w _m are given real numbers. In particular, if this dimension ism, i.e. ifw ₁,...,w _m are linearly independent over , we are also interested in a quantitative version of this fact. Our qualitative theorems generalize a result of Nesterenko. Its formulation is quite similar to the axiomatization of methods for algebraic independence, as it became usual during the last decade.

     

Impossibility of a Certain Cyclotomic Equation with Applicationsto Difference Sets

Under certain conditions, we show the nonexistence ofan element in the p-th cyclotomicfield over , that satisfies . As applications, we establish the nonexistence ofsome difference sets and affine difference sets.     

Categories equivalent to the category of rational H-spaces

The category of rationalH-spaces is shown to be equivalent to the category of commutative Hopf algebras over , the category of cocommutative Hopf algebras over , and the categoryL of graded Lie algebras over by the rational cohomology, homology, and homotopy functors, respectively. Several consequences of these equivalences are derived. It is also proved that the loop-space functor is an equivalence from the category of coformal rational spaces to . Dually, the category of rationalcoH-spaces is shown to be equivalent to the comonoid category ofL and to the category of cocommutative coalgebras over . The suspension functor is an equivalence from the category of formal, rational spaces to the category of 2-connected, rationalcoH-spaces.     

Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

Abelianness of Mumford–Tate Groups Associated to Some Unitary Groups

In this paper, we investigate the action of the -cohomology of the compact dual of a compact Shimura Variety S() on the -cohomology of S()> under a cup product. We use this to split the cohomology of S() into a direct sum of (not necessarily irreducible) -Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on S() are Abelian. As another application, we show that all classes of Hodge type (1,1) in H² of unitary four-folds associated to the group U(2,2) are algebraic.     

Note on initial topologies on rational vector spaces induced by realvalued linear mappings

LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by _f , _g the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf ^–1 U _g , then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G, _f )(G, _g ) is continuous if and only if there is a real constantc withg ^o h=cf.Dedicated to Professor János Aczél on his 60th birthday     

Complexity of deciding the first-order theory of real closed fields

Let denote an ordered field, where _i+1 is infinitesimal relative to the elements of the field _i 0 i < m (by definition, ₀=). Given a formula of the first-order theory of the real closed field _m, of the following form:, where P is a quantifier-free formula containing k atomic subformulas of the form (f_j 0), where are polynomials such that, and the absolute value of any integer occurring in the coefficients of the polynomials f_j is at most 2_M. Let =S₁+...+S_a denote the number of variables anda n the number of quantifiers in the formula. An algorithm is constructed which decides the truth of formulas of the above form in polynomial time with respect to M,. Previously known algorithms for this problem had complexity of the order of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 53–100, 1988.     

Über Hyperflächen aus Quadriken und quadratischen Kegeln im reellenm-dimensionalen projektiven Raum

Zusammenfassung Die vorgestellte Kennzeichnung für eine reguläreC ²(U)-Hyperflächef(U) mit zusammenhängender offener ParametermengeU ^m–1 aus einer Quadrik im reellenm-dimensionalen projektiven Raum ^m ,m 3 über dem (m+l)-dimensionalen reellen Vektorraum ^m+1 stützt sich auf die ebenen Schnitte vonf(U) durch eine im Punktf(a) der Hyperfläche nichttangentiale Geradeg. Als Anwendung ergeben sich einerseits eine Charakterisierung von Hyperflächen aus einer Quadrik unter Benützung gewisser scheinbarer Umrisse und andererseits eine Verallgemeinerung eines Ergebnisses über die zentralaffine Metrik nach D. Laugwitz. Weiters werden Hyperflächen aus quadratischen Kegeln in ^m mit Hilfe ihrer Schnitte mit Unterräumen geeigneter fester Dimension durch eine Gerade gekennzeichnet.Wir verwenden wie in [6] einen Flächenbegriff unter Berücksichtigung der Quotientenstruktur von ^m im Sinne differenzierbarer Mannigfaltigkeiten. Unter Benützung von Aussagen aus [6] wird für reguläre Hyperflächen in ^m die Existenz entweder einer oskulierenden Quadrik oder eines oskulierenden quadratischen Kegels oder einer oskulierenden Hyperebene in jedem Flächenpunkt nachgewiesen.     

Semistable abelian varieties over ℚ

We prove that for N=6 and N=10, there do not exist any non-zero semistable abelian varieties over with good reduction outside primes dividing N. Our results are contingent on the GRH discriminant bounds of Odlyzko. Combined with recent results of Brumer-Kramer and of Schoof, this result is best possible: if N is squarefree, there exists a non-zero semistable abelian variety over with good reduction outside primes dividing N precisely when N{1,2,3,5,6,7,10,13}.Mathematics Subject Classification (2000): 14K15     

On Conjectures of Mathai and Borel

Mathai has conjectured that the Cheeger–Gromov invariant ₍₂₎ = ₍₂₎ - is a homotopy invariant of closed manifolds with torsion-free fundamental group. In this paper we prove this statement for closed manifolds M when the rational Borel conjecture is known for = ₁(M), i.e. the assembly map : H _*(B, ) L_*() is an isomorphism. Our discussion evokes the theory of intersection homology and results related to the higher signature problem.     

Sur la construction d'un corps de Hilbert

We are concerned with the following: If k is a quadratic field and N a cyclic unramified extension of degree qⁿ over k, q a prime number, determine N explicitely via a primitive element , i.e., N=k(), in the spirit of Helmut Hasse [3]. We propose a method which determines these extensions, once we are able to specify the arithmetic of a certain field . To explicit our method, we construct the Hilbert fields of (226) and (646).     

On finite and totally finite elements in vector lattices

, X , , , M l- M X , ( M) M .     

λ-invariants and Γ-transforms

In this paper we give the relationship between the -invariants of two power series which are naturally associated to the -transform of ap-adic measure. Since thep-adicL-functions over arise as such -transforms, we obtain information about the minus part of Iwasawa's -invariant for the basic _p -extension of an abelian CM-field.I would like to thank W. Sinnott, R. Gold and T. Dowling for their comments and suggestions.