Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

Calculation of values of -functions associated to elliptic curves

We calculated numerically the values of -functions of four typical elliptic curves in the critical strip in the range . We found that all the non-trivial zeros in this range lie on the critical line and are simple except the one at . The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane , which enables us to calculate for large . Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

Nonvanishing of symmetric square -functions of cusp forms inside the critical strip

We shall give a certain nonvanishing result for the symmetric square -function of an elliptic cuspidal Hecke eigenform w.r.t. the full modular group inside the critical strip.

     

coefficients

Let be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain (). We consider the differential operator

where the coefficients are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

for a.e. , every , some constant . Moreover, we assume that the coefficients belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields . We prove the following local -estimate:

for every , . We also prove the local Hölder continuity for solutions to for any with large enough. Finally, we prove -estimates for higher order derivatives of , whenever and the coefficients are more regular.

     

Willmore Functionals

We prove some integral inequalities for immersed tori in the three sphere. The functionals considered are generalizations of the Willmore functional.

     

Products on -modules

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of -modules such as , and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over that are concentrated in degrees divisible by ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising -local -modules as -modules.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Metrizable and -metrizable betweenness spaces

It is proved that the theory of the class of all betweenness spaces metrizable by real-valued metrics does not coincide with the theory of the class of all betweenness spaces metrizable by metrics taking values in any ordered field. This solves a problem raised by Mendris and Zlatov{s}.

     

A counterexample machine

We present a general method for constructing families of measure preserving transformations which are and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two transformations which are weakly isomorphic but not isomorphic, and a transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.

     

Determining the small solutions to -unit equations

In this paper we generalize the method of Wildanger for finding small solutions to unit equations to the case of -unit equations. The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm. The method used reduces the computing time needed from MIPS years down to minutes.

     

A generalization of Kelley's theorem for -spaces

We prove that if an open map of compacta and has perfect fibers and is a -space, then there exists a -dimensional compact subset of intersecting each fiber of . This is a stronger version of a well-known theorem of Kelley. Applications of this result and related topics are discussed.

     

The -theory of toric manifolds

Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an -dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the -theory of all toric manifolds and certain singular toric varieties.

     

Nonexistence conditions of a solution for the congruence

We obtain nonexistence conditions of a solution for of the congruence , where , and are integers, and is a prime power. We give nonexistence conditions of the form for , , , , , and of the form for , , , . Furthermore, we complete some tables concerned with Waring's problem in -adic fields that were computed by Hardy and Littlewood.

     

The covering numbers and ``low -estimate" for quasi-convex bodies

This article gives estimates on the covering numbers and diameters of random proportional sections and projections of quasi-convex bodies in . These results were known for the convex case and played an essential role in the development of the theory. Because duality relations cannot be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.

     

Operator ideal norms on

Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

     

The length of -pseudodifferential operators

We compute the length of the -algebra generated by the algebra of b-pseudodifferential operators of order on compact manifolds with corners.

     

estimates for oscillatory integral operators

An endpoint boundedness result is established for a class of oscillatory integral operators.