Abelian subgroups of pro- Galois groups

It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois -extension of a field (where is an odd prime) arise from -henselian valuations with non--divisible value group, provided and contains a primitive -th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.

     

A -deformation of a trivial symmetric group action

Let be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system . Let be the Varchenko matrix for this arrangement with all hyperplane parameters equal to . We show that is the matrix with rows and columns indexed by permutations with entry equal to where is the number of inversions of . Equivalently is the matrix for left multiplication on by

Clearly commutes with the right-regular action of on . A general theorem of Varchenko applied in this special case shows that is singular exactly when is a root of for some between and . In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the -module structure of the nullspace of in the case that is singular. Our first result is that

in the case that where Lie denotes the multilinear part of the free Lie algebra with generators. Our second result gives an elegant formula for the determinant of restricted to the virtual -module with characteristic the power sum symmetric function .

     

Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Let denote the classical equilibrium distribution (of total charge ) on a convex or -smooth conductor in with nonempty interior. Also, let be any th order ``Fekete equilibrium distribution' on , defined by point charges at th order ``Fekete points'. (By definition such a distribution minimizes the energy for -tuples of point charges on .) We measure the approximation to by for by estimating the differences in potentials and fields,

both inside and outside the conductor . For dimension we obtain uniform estimates at distance from the outer boundary of . Observe that throughout the interior of (Faraday cage phenomenon of electrostatics), hence on the compact subsets of . For the exterior of the precise results are obtained by comparison of potentials and energies. Admissible sets have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions of equal point charges on . In that context there is an important open problem for the sphere which is discussed at the end of the paper.

     

Tauberian theorems and stability of solutions of the Cauchy problem

Let be a bounded, strongly measurable function with values in a Banach space , and let be the singular set of the Laplace transform in . Suppose that is countable and uniformly for , as , for each in . It is shown that

as , for each in ; in particular, if is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on , and it implies several results concerning stability of solutions of Cauchy problems.

     

An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.

     

Comultiplications on free groups and wedges of circles

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group . We consider individual comultiplications on and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of . For a comultiplication of we define a subset of quasi-diagonal elements which is basic to our investigation of associativity. The subset can be determined algorithmically and contains the set of diagonal elements . We show that is a basis for the largest subgroup of on which is associative and that is a free factor of . We also give necessary and sufficient conditions for a comultiplication on to be a coloop in terms of the Fox derivatives of with respect to a basis of . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group on the set of comultiplications of . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

Rumely's local global principle for algebraic PC fields over rings

Let be a finite set of rational primes. We denote the maximal Galois extension of in which all totally decompose by . We also denote the fixed field in of elements in the absolute Galois group of by . We denote the ring of integers of a given algebraic extension of by . We also denote the set of all valuations of (resp., which lie over ) by (resp., ). If , then denotes the ring of integers of a Henselization of with respect to . We prove that for almost all , the field satisfies the following local global principle: Let be an affine absolutely irreducible variety defined over . Suppose that for each and for each . Then . We also prove two approximation theorems for .

     

Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials

We prove that for every rational map on the Riemann sphere , if for every -critical point whose forward trajectory does not contain any other critical point, the growth of is at least of order for an appropriate constant as , then . Here is the so-called essential, dynamical or hyperbolic dimension, is Hausdorff dimension of and is the minimal exponent for conformal measures on . If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of also coincides with . We prove ergodicity of every -conformal measure on assuming has one critical point , no parabolic, and . Finally for every -conformal measure on (satisfying an additional assumption), assuming an exponential growth of , we prove the existence of a probability absolutely continuous with respect to , -invariant measure. In the Appendix we prove also for every non-renormalizable quadratic polynomial with not in the main cardioid in the Mandelbrot set.

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .

     

Geometry of families of nodal curves on the blown-up projective plane

Let be the projective plane blown up at generic points. Denote by the strict transform of a generic straight line on and the exceptional divisors of the blown-up points on respectively. We consider the variety of all irreducible curves in with nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For we give the complete answer concerning the existence of nodal curves in .

     

Values of Gaussian hypergeometric series

Let be prime and let be the finite field with elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions

where and respectively are the quadratic and trivial characters of For all but finitely many rational numbers there exist two elliptic curves and for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes for which is zero; however if or , then the set of such primes has density zero. In contrast, if or , then there are only finitely many primes for which Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating over every

     

Singular limit of solutions of the porous medium equation with absorption

We prove that as the solutions of , , , , , , , converges in to the solution of the ODE , , where , , satisfies in for some function , , satisfying whenever for a.e. , for and for , where is a constant and is any measurable subset of .

     

Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map

Let be a locally compact group and the Fourier-Stieltjes algebra of . We study the problem of how weak*-closedness of some translation invariant subspaces of is related to the structure of . Moreover, we prove that for a closed subgroup of , the restriction map from to is weak*-continuous only when is open in .

     

On zeta functions and Iwasawa modules

We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules for almost all determine the zeta function when is a totally real field. Conversely, we prove that the -part of the Iwasawa module is determined by its zeta-function up to pseudo-isomorphism for any number field Moreover, we prove that arithmetically equivalent CM fields have also the same -invariant.

     

The Stable Homotopy Types of Stunted Lens Spaces mod

Let be the mod stunted lens space . Let denote the exponent of in , and the number of integers satisfying , and . In this paper we complete the classification of the stable homotopy types of mod stunted lens spaces. The main result (Theorem 1.3 (i)) is that, under some appropriate conditions, and are stably equivalent iff , where or .

     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

Emergence of the Witt group in the cellular lattice of rational spaces

We give an embedding of a quotient of the Witt semigroup into the lattice of rational cellular classes represented by formal -cones between and the two-cell complex ().