The arithmetic and combinatorics of buildings for

In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for with a local field. We characterize the action of the affine Weyl group in terms of a symplectic basis for an apartment, characterize the special vertices as those which are self-dual with respect to the induced inner product, and establish a one-to-one correspondence between the special vertices in an apartment and the elements of the quotient .

We then give a natural representation of the local Hecke algebra over acting on the special vertices of the Bruhat-Tits building for . Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for .

     

Square -lattice paths and

The combinatorial -Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The -Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the 'th -Catalan number is the Hilbert series for the module of diagonal harmonic alternants in variables; it is also the coefficient of in the Schur expansion of . Using -analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of and the Hilbert series of the diagonal harmonics modules.

This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several -analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the -Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of , the ``Hilbert series' , and the sign character .

     

of Hamiltonian manifolds

Let be a connected, compact symplectic manifold equipped with a Hamiltonian action. We prove that, as fundamental groups of topological spaces, , where is the symplectic quotient at any value in the image of the moment map .

     

Compatible valuations and generalized Milnor -theory

Given a field and a subgroup of there is a minimal group for which there exists an -compatible valuation whose units are contained in . Assuming that has finite index in and contains for prime, we describe in computable -theoretic terms.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

On -adic intermediate Jacobians

For an algebraic variety of dimension with totally degenerate reduction over a -adic field (definition recalled below) and an integer with , we define a rigid analytic torus together with an Abel-Jacobi mapping to it from the Chow group of codimension algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over . We compare and contrast the complex and -adic theories. Finally, we examine a special case of a -adic analogue of the Generalized Hodge Conjecture.

     

-continuity properties of the symmetric -stable process

Let be a domain of finite Lebesgue measure in and let be the symmetric -stable process killed upon exiting . Each element of the set of eigenvalues associated to , regarded as a function of , is right continuous. In addition, if is Lipschitz and bounded, then each is continuous in and the set of associated eigenfunctions is precompact.

     

Uniformizable families of -motives

Abelian -modules and the dual notion of -motives were introduced by Anderson as a generalization of Drinfeld modules. For such Anderson defined and studied the important concept of uniformizability. It is an interesting question and the main objective of the present article to see how uniformizability behaves in families. Since uniformizability is an analytic notion, we have to work with families over a rigid analytic base. We provide many basic results, and in fact a large part of this article concentrates on laying foundations for studying the above question. Building on these, we obtain a generalization of a uniformizability criterion of Anderson and, among other things, we establish that the locus of uniformizability is Berkovich open.

     

Legendrian contact homology in

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form , where is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of and, more generally, invariants of self transverse immersions into up to restricted regular homotopies. When , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

     

The -stable pieces of the wonderful compactification

Let be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification of into finite many -stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any -stable piece in . We will show that the closure is a disjoint union of some -stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many -orbits.

     

Extensions of -local finite groups

A -local finite group consists of a finite -group , together with a pair of categories which encode ``conjugacy' relations among subgroups of , and which are modelled on the fusion in a Sylow -subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as -completed classifying spaces of finite groups. In this paper, we study and classify extensions of -local finite groups, and also compute the fundamental group of the classifying space of a -local finite group.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

Iwasawa theory for -local spectra

The Iwasawa algebra is a power series ring in one variable over the -adic integers. It has long been studied by number theorists in the context of -extensions of number fields. It also arises, however, as a ring of operations in -adic topological -theory. In this paper we study -local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for odd we classify -local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for -modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.

     

estimates for overdetermined Radon transforms

We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning bounds for convolution with all rotations of arc length measure on a fixed convex curve in . Estimates are obtained for averages over higher-dimensional convex (nonsmooth) hypersurfaces, smooth -dimensional surfaces, and nontranslation-invariant families of surfaces. We compare Ricci and Travaglini's approach, based on average decay of the Fourier transform, with an approach based on boundedness of Fourier integral operators, and show that essentially the same geometric condition arises in proofs using the two techniques.

     

-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.

     

All -local finite groups of rank two for odd prime

In this paper we give a classification of the rank two -local finite groups for odd . This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible -radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic -local finite group, getting some new examples of these for .

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

On algebraic -groups

We introduce the categories of algebraic -varieties and -groups over a difference field . Under a ``linearly -closed" assumption on we prove an isotriviality theorem for -groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski's work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).