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.     

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.     

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.     

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).     

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).     

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.     

Hecke algebras,type III factors and phase transitions with spontaneous symmetry breaking in number theory

In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings provides the desiredC*-dynamical system, which admits the function as partition function and the Galois group Gal(^cycl/) of the cyclotomic extension ^cycl of as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature =1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).     

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.

     

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 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.     

Die stabile Reduktion der Fermatkurve über einem Zahlkörper

Let p be an odd prime and F the Fermat curve of degree p, defined by x^p+y^p=1 over . Although the curve F has bad reduction at the prime (p), the stable reduction theorem assures that over some number field K/ we can get stable reduction of the curve F at the primes lying above p. We have determined it in this paper. See Abb.1.     

Perron-frobenius theory for Banach spaces with a hyperbolic cone

If X is a real Banach space, then the inequality x defines so-called hyperbolic cone in E=X. We develop a relevant version of Perron-Frobenius-Krein-Rutman theory.     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

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.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

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.     

Fields with vanishing K2. Torsion in H1(X,K2) and Ch2(x)

This paper describes fields F of nonzero characteristic with the property that for all finite extensions E/F K₂E=0. We consider a somewhat wider class of fields which includes finite and separably closed fields. For smooth projective varieties X over such a field we show that the groups H¹(X, K₂){} and H²(X_et, (2)), NH³(X_et, (2)) and Ch²(X){} are isomorphic. These results are applied to describe the groups SK₁ of a smooth affine curve over such a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 108–118, 1982.