Computing

Let denote the Von Mangoldt function and . We describe an elementary method for computing isolated values of . The complexity of the algorithm is time and space. A table of values of for up to is included, and some times of computation are given.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Involutions with

Let be a smooth involution on a closed -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each component of the fixed point set of vanish in positive dimension. In this paper, we estimate the least possible lower bound of dim if does not bound.

     

Projections from

Let and be two infinite dimensional real Banach spaces. The following question is classical and long-standing. Are the following properties equivalent?

a) There exists a projection from the space of continuous linear operators onto the space of compact linear operators.

b) .

The answer is positive in certain cases, in particular if or has an unconditional basis. It seems that there are few results in the direction of a general solution. For example, suppose that and are reflexive and that or has the approximation property. Then, if , there is no projection of norm 1, from onto . In this paper, one obtains, in particular, the following result:

Theorem. Let be a real Banach space which is reflexive (resp. with a separable dual), of infinite dimension, and such that has the approximation property. Let be a real scalar with . Then can be equivalently renormed such that, for any projection from onto , one has . One gives also various results with two spaces and .

     

On the Normal Subgroups of

We give a characterization theorem for the -normalized subgroups of , where is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking.

     

Crystal bases for

A construction of the crystal bases for the quantized enveloping algebra of is discussed.

     

Selective ultrafilters and

Mathias (Happy families, Ann. Math. Logic. 12 (1977), 59-111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in is -Ramsey with respect to every selective ultrafilter . In this paper, we show that the large cardinal assumption cannot be weakened.

     

Shintani functions on

In this paper, we give a formulation and an explicit formula for Shintani function on , which has been studied by Murase and Sugano in the theory of automorphic -functions. In particular, we obtain the multiplicity of this function.

     

On the unitary dual of

In this paper we begin a systematic study of the unitarity question for genuine representations of the group . The main result is that we find a class of unitary representations (mostly isolated) analogous to the special unipotent representations defined by D. Barbasch and D. Vogan. In particular the full unitary dual of should be obtainable from this set by complementary series.

     

Left-symmetric algebras for

We study the classification problem for left-symmetric algebras with commutation Lie algebra in characteristic . The problem is equivalent to the classification of étale affine representations of . Algebraic invariant theory is used to characterize those modules for the algebraic group which belong to affine étale representations of . From the classification of these modules we obtain the solution of the classification problem for . As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.

     

On 2-Generator Subgroups of

We classify all subgroups of that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of . In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by and . Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.

     

The Diophantine equation

If and are given positive integers with , then we show that the equation of the title possesses at most one solution in positive integers . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.

     

groups and quasi-equivalence

A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

     

Conjugacy Classes of

Let be a quadratic extension of a global field , of characteristic not two, and the integral closure in of a Dedekind ring of -integers in . Then is isomorphic to the spinorial kernel for an indefinite quadratic -lattice of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups of primitive odd binary hermitian matrices under the action of .

     

Monomial Bases for

We consider the polynomial algebra as a module over the mod Steenrod algebra, , being an odd prime. We give a minimal set of generators consisting of monomials and characterise all such `monomial bases'.

     

Paranormal spaces under

We prove that paranormal spaces of character are -
collectionwise Hausdorff assuming the set-theoretic principle . This gives an affirmative answer to problem 197 in Problems I wish I could solve, by W. S. Watson (Open Problems in Topology (1990), 37-76).

     

Spin L-functions on

The ``spin' L-function of an automorphic representation of is an Euler product of degree associated with the spin representation of the L-group . If or , and the automorphic representation is generic in the sense of having a Whittaker model, the analytic properties of these L-functions are studied by the Rankin-Selberg method.