On the nonvanishing of the central value of the Rankin-Selberg -functions

We characterize the nonvanishing of the central value of the Rankin-Selberg -functions in terms of periods of Fourier-Jacobi type. This characterization is based on the Langlands philosophy on functoriality and on applications of invariant theory to automorphic representations. The result is the symplectic analog of the Gross-Prasad conjecture.

     

Small unitary representations of the double cover of

The irreducible unitary representations of the double cover of the real group , with infinitesimal character , which are small in the sense that their annihilator in the universal enveloping algebra is maximal, are expressed as Langlands quotients of generalized principal series. In the case where is even we show that there are four such representations and in the case where is odd there is just one. The representations' smallness allows them to be written as a sum of virtual representations, leading to a character formula for their -types. We investigate the place of these small representations in the orbit method and, in the case of , show that the representation is attached to a nilpotent coadjoint orbit.The -type spectrum for the Langlands quotients is explicitly determined and shown to be multiplicity free.

     

-theory of -pseudo-differential algebras

We are concerned with the so-called -pseudo-differential calculus. We describe the spectrum of the unital and commutative -algebra given by the norm closure of the space of -order pseudo-differential operators modulo compact operators; other related algebras are also considered. Finally, their -theory is computed.

     

Computing weight modular forms of level

For a prime we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space of modular forms of weight and level . For we define a special Hecke stable subspace of which contains the space of modular forms with CM by the ring of integers of and we describe the calculation of the corresponding Brandt matrices.

     

Cut numbers of -manifolds

We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

     

The distribution functions of and

Let be the sum of the positive divisors of . We show that the natural density of the set of integers satisfying is given by , where denotes Euler's constant. The same result holds when is replaced by , where is Euler's totient function.

     

A hereditarily subspace of without the Schur property

Let p_1 > p_2 > \cdots > 1$">. We construct an easily determined -symmetric basic sequence in , which spans a hereditarily subspace without the Schur property. An immediate consequence is the existence of hereditarily subspaces of without the Schur property.

     

-actions with

Suppose that is a smooth -action on a closed smooth -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set vanish in positive dimension. This paper shows that if 2^k\dim F$"> and each -dimensional part possesses the linear independence property, then bounds equivariantly, and in particular, is the best possible upper bound of if is nonbounding.

     

-inner automorphisms of finite groups

We refer to an automorphism of a group as -inner if given any subset of with cardinality less than , there exists an inner automorphism of agreeing with on . Hence is 2-inner if it sends every element of to a conjugate. New examples are given of outer -inner automorphisms of finite groups for all natural numbers .

     

The -exponent of the -local spectrum

Let be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the -exponent of the spectrum is for . It follows from this result that the -exponent of is at least for and , where denotes the -connected cover of .

     

Twisted sums with spaces

If is a separable Banach space, we consider the existence of non-trivial twisted sums , where or For the case we show that there exists a twisted sum whose quotient map is strictly singular if and only if contains no copy of . If we prove an analogue of a theorem of Johnson and Zippin (for ) by showing that all such twisted sums are trivial if is the dual of a space with summable Szlenk index (e.g., could be Tsirelson's space); a converse is established under the assumption that has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with with strictly singular quotient map.

     

Contact structures on elliptic -manifolds

We show that an oriented elliptic -manifold admits a universally tight positive contact structure if and only if the corresponding group of deck transformations on (after possibly conjugating by an isometry) preserves the standard contact structure.

We also relate universally tight contact structures on -manifolds covered by to the isomorphism .

The main tool used is equivariant framings of -manifolds.

     

Modular group algebras of -separable -groups

Under the assumptions of MA and CH, it is proved that if is a field of prime characteristic and is an -separable abelian -group of cardinality , then an isomorphism of the group algebras and implies an isomorphism of and .

     

Whittaker models for , a -adic field

Let be a non-archimedean local field with odd residual characteristic. Let be the group We construct explicit Whittaker models for any supercuspidal representation of with positive level.

     

-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.

     

-bounding and -induction

Working in the base theory of , we show that for all , the bounding principle for -formulas ( ) is equivalent to the induction principle for -formulas ( ). This partially answers a question of J. Paris.

     

The threshold for random -SAT is

Let be a random -CNF formula formed by selecting uniformly and independently out of all possible -clauses on variables. It is well known that if , then is unsatisfiable with probability that tends to 1 as . We prove that if , where , then is satisfiable with probability that tends to 1 as .

Our technique, in fact, yields an explicit lower bound for the random -SAT threshold for every . For our bounds improve all previously known such bounds.

     

Extending into isometries of

In this paper we generalize a result of Hopenwasser and Plastiras (1997) that gives a geometric condition under which into isometries from to have a unique extension to an isometry in . We show that when and are separable reflexive Banach spaces having the metric approximation property with strictly convex and smooth and such that is a Hahn-Banach smooth subspace of , any nice into isometry has a unique extension to an isometry in .

     

Interpolation between and

We show that if is a rearrangement invariant space on that is an interpolation space between and and for which we have only a one-sided estimate of the Boyd index 1/p, 1 < p < \infty$">, then is an interpolation space between and . This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type and weak type .

     

Pseudocompact spaces and -spaces

Let be a completely regular Hausdorff space, and let be the space of continuous real-valued functions on endowed with the compact-open topology. We find various equivalent conditions for to be a -space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for , Grothendieck's -spaces do not coincide with Jarchow's -spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for , our original motivation.