Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

Monomial bases for -Schur algebras

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of and its associated monomial basis, we investigate -Schur algebras as ``little quantum groups". We give a presentation for and obtain a new basis for the integral -Schur algebra , which consists of certain monomials in the original generators. Finally, when , we interpret the Hecke algebra part of the monomial basis for in terms of Kazhdan-Lusztig basis elements.

     

Hyperbolic -spheres with conical singularities, accessory parameters and Kähler metrics on

We show that the real-valued function on the moduli space of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic -sphere with conical singularities of arbitrary orders , generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on parameterized by the set of orders , explicitly relate accessory parameters to these metrics, and prove that the functions are their Kähler potentials.

     

Estimations des fonctions du Laplacien sur les variétés cuspidales

Le but de cet article est d'étudier la continuité des fonctions du Laplacien sur les variétés cuspidales.

     

Formulas for tamely ramified supercuspidal characters of

Let denote a -adic local field of residual characteristic . This article gives formulas, valid on the regular elliptic set, for the irreducible supercuspidal characters of which correspond to characters of a ramified Cartan subgroup. In the case in which does not contain cube roots of unity, i.e., the case in which ramified cubic extensions of degree over cannot be Galois, base change results concerning ``simple types" due to Bushnell and Henniart (1996) are used in the proofs.

     

-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.

     

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 .

     

Constructive recognition of

Existing black box and other algorithms for explicitly recognising groups of Lie type over have asymptotic running times which are polynomial in , whereas the input size involves only . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises explicitly.

The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising in its natural representation in polynomial time, given a discrete logarithm oracle for . The algorithm presented here takes as input a generating set for a subgroup of that is isomorphic modulo scalars to , where is a finite field of the same characteristic as ; it returns the natural representation of modulo scalars. Since a faithful projective representation of in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in rather than in , elementary algorithms will recognise explicitly in polynomial time in these cases. Given a discrete logarithm oracle for , our algorithm thus provides the required polynomial time oracle for recognising explicitly in the remaining case, namely for representations in the natural characteristic.

This leads to a partial solution of a question posed by Babai and Shalev: if is a matrix group in characteristic , determine in polynomial time whether or not is trivial.

     

On the mod cohomology of

We study the mod  cohomology of the classifying space of the projective unitary group . We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod  cohomology of the classifying space of connected compact Lie groups hold in the case of . Finally, we prove that the classifying space of the projective unitary group is determined by its mod  cohomology as an unstable algebra over the Steenrod algebra for 3$">, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases .

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

On the -compact groups corresponding to the -adic reflection groups

There exists an infinite family of -compact groups whose Weyl groups correspond to the finite -adic pseudoreflection groups of family 2a in the Clark-Ewing list. In this paper we study these -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical -homomorphism. Finally, we also describe a faithful complexification homomorphism from these -compact groups to the -completion of unitary compact Lie groups.

     

On embedding all -manifolds into a single -manifold

For each composite number , there does not exist a single connected closed -manifold such that any smooth, simply-connected, closed -manifold can be topologically flatly embedded into it. There is a single connected closed -manifold such that any simply-connected, -manifold can be topologically flatly embedded into if is either closed and indefinite, or compact and with non-empty boundary.

     

Singular integrals with rough kernels along real-analytic submanifolds in

mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translates of a real-analytic submanifold in .

     

A function for which the -limit points are not contained in the closure of the periodic points

We develop a function for which . This answers a query from Block and Coppel (1992).

     

Blakers-Massey elements and exotic diffeomorphisms of and via geodesics

We use the geometry of the geodesics of a certain left-invariant metric on the Lie group to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of ) and an exotic (i.e. not isotopic to the identity) diffeomorphism of (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of and exotic diffeomorphisms of , thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.

     

Associahedra, cellular -construction and products of -algebras

The aim of this paper is to construct a functorial tensor product of -algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.

     

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 .

     

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 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).