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.     

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.     

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.     

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.

     

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.     

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.     

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 .     

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).     

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]).     

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.     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.     

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).     

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.     

A Measure of Simultaneous Approximation for Quasi-Modular Functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient of a period of an elliptic curve defined over by its associated quasi-period. In an (almost successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the modular method ideas related to modular but also hypergeometric functions, as appearing e.g. in André's work, as well as some Galois considerations.     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

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Some functionals over a compact Minkovskii space

In this paper we obtain estimates which are order-exact for the projection and Macphail constants of an arbitrary n-dimensional Banach space: 1(X)n, 1/n₁(X)1/n.Translated from Matematicheskii Zametki, Vol. 10, No. 4, pp. 453–457, 1971.