Semi-linear homology -spheres and their equivariant inertia groups

This paper introduces an abelian group for all semi-linear homology -spheres, which corresponds to a known abelian group for all semi-linear homotopy -spheres, where is a compact Lie group and is a -representation with 0$">. Then using equivariant surgery techniques, we study the relation between both and when is finite. The main result is that under the conditions that -action is semi-free and with 0$">, the homomorphism defined by is an isomorphism if , and a monomorphism if . This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology -spheres.

     

Cartan-decomposition subgroups of

For and , we give explicit, practical conditions that determine whether or not a closed, connected subgroup of has the property that there exists a compact subset of with . To do this, we fix a Cartan decomposition of , and then carry out an approximate calculation of for each closed, connected subgroup of .

     

How to do a -descent on an elliptic curve

In this paper, we describe an algorithm that reduces the computation of the (full) -Selmer group of an elliptic curve over a number field to standard number field computations such as determining the (-torsion of) the -class group and a basis of the -units modulo th powers for a suitable set of primes. In particular, we give a result reducing this set of `bad primes' to a very small set, which in many cases only contains the primes above . As of today, this provides a feasible algorithm for performing a full -descent on an elliptic curve over , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of is favorable, simplifications are possible and -descents for larger are accessible even today. To demonstrate how the method works, several worked examples are included.

     

The cohomology of certain Hopf algebras associated with -groups

We study the cohomology of a locally finite, connected, cocommutative Hopf algebra over . Specifically, we are interested in those algebras for which is generated as an algebra by and . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras with monogenic and semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for to be semi-Koszul. Special attention is given to the case in which is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- lower central series of a -group. We show that the algebras arising in this way from extensions by of an abelian -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 -groups, and it is shown that these are all semi-Koszul for .

     

Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on

We investigate the relationship between the decay at infinity of the right-hand side and solutions of an equation when is a second order elliptic operator on It is shown that when is Fredholm, inherits the type of decay of (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.

     

Exponential sums on , II

We prove a vanishing theorem for the -adic cohomology of exponential sums on . In particular, we obtain new classes of exponential sums on that have a single nonvanishing -adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

     

A unified approach to improved Hardy inequalities with best constants

We present a unified approach to improved Hardy inequalities in . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension . In our main result, we add to the right hand side of the classical Hardy inequality a weighted norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted norms, .

     

On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

Parametrized principles

We will present a collection of guessing principles which have a similar relationship to as cardinal invariants of the continuum have to . The purpose is to provide a means for systematically analyzing and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of and in models such as those of Laver, Miller, and Sacks.

     

Involutions fixing , II

This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space with its normal bundle nonbounding and a Dold manifold with a positive even and 0$">. The complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on the codimension of may not be best possible. In particular, we find that there exist such involutions with nonstandard normal bundle to . Together with the results of part I of this title (Trans. Amer. Math. Soc. 354 (2002), 4539-4570), the argument for involutions fixing is finished.

     

Morse index and uniqueness for positive solutions of radial -Laplace equations

We study the positive radial solutions of the Dirichlet problem in , 0$"> in , on , where , 1$">, is the -Laplace operator, is the unit ball in centered at the origin and is a function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of . We use this to prove uniqueness and nondegeneracy of positive radial solutions when is of the type and .

     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

Lattice invariants and the center of the generic division ring

Let be a finite group, let be a -lattice, and let be a field of characteristic zero containing primitive roots of 1. Let be the quotient field of the group algebra of the abelian group . It is well known that if is quasi-permutation and -faithful, then is stably equivalent to . Let be the center of the division ring of generic matrices over . Let be the symmetric group on symbols. Let be a prime. We show that there exist a split group extension of by a -elementary group, a -faithful quasi-permutation -lattice , and a one-cocycle in such that is stably isomorphic to . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if is algebraically closed, there is a group extension of by an abelian -group such that is stably equivalent to the invariants of the Noether setting .

     

Positive laws in fixed points

Let be an elementary abelian group of order at least acting on a finite -group in such a manner that satisfies a positive law of degree for any . It is proved that the entire group satisfies a positive law of degree bounded by a function of and only.

     

Combinatorial properties of Thompson's group

We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group in the standard two generator presentation. We explore connections between the tree pair diagram representing an element of , its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of with respect to the two generator finite presentation. Namely, we exhibit the form of ``dead end' elements in this Cayley graph, and show that it has no ``deep pockets'. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.

     

On the -Minkowski problem

A volume-normalized formulation of the -Minkowski problem is presented. This formulation has the advantage that a solution is possible for all , including the degenerate case where the index is equal to the dimension of the ambient space. A new approach to the -Minkowski problem is presented, which solves the volume-normalized formulation for even data and all .

     

Ideals of the cohomology rings of Hilbert schemes and their applications

We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

     

Rationality, regularity, and -cofiniteness

We demonstrate that, for vertex operator algebras of CFT type, -cofiniteness and rationality is equivalent to regularity. For -cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and -cofinite, is -cofinite, and the fusion rules are finite.

     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.