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.

     

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 .

     

On one-dimensional self-similar tilings and -tiles

Let be an integer base, a digit set and the set of radix expansions. It is well known that if has nonvoid interior, then can tile with some translation set ( is called a tile and a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of ; (ii) for a given , characterize so that is a tile.

We show that for a given pair , there is a unique self-replicating translation set , and it has period for some . This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for when are distinct primes. The only other known characterization is for , due to Lagarias and Wang. The proof for the case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.

     

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.

     

Resolutions of ideals of quasiuniform fat point subschemes of

The notion of a quasiuniform fat point subscheme is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal defining are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the th symbolic power of an ideal defining general points of when both and are large (in particular, for infinitely many for each of infinitely many , and for infinitely many for every 2$">). Resolutions in other cases, such as ``fat points with tails', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field . As an incidental result, a bound for the regularity of is given which is often a significant improvement on previously known bounds.

     

The space for nondoubling measures in terms of a grand maximal operator

Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , as in the usual doubling situation.

     

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.

     

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.

     

On the minimal free resolution of general forms

Let and let be the ideal of generically chosen forms of degrees . We give the precise graded Betti numbers of in the following cases:

• ;
• and is even;
• , is odd and ;
• is even and all generators have the same degree, , which is even;
• is even and ;
• is odd, is even, and .

We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e., one for which the Hilbert function is maximal, given and the socle degree) when is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree, then there are no redundant summands, and we present some evidence for this conjecture.

     

The Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem

for two second-order elliptic operators , , in a bounded Lipschitz domain . The coefficients belong to the space of bounded mean oscillation with a suitable small modulus. We assume that is regular in for some , , that is, for all continuous boundary data . Here is the surface measure on and is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients that will assure the perturbed operator to be regular in for some , .

     

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.

     

Explicit Lower bounds for residues at of Dedekind zeta functions and relative class numbers of CM-fields

Let be a given set of positive rational primes. Assume that the value of the Dedekind zeta function of a number field is less than or equal to zero at some real point in the range . We give explicit lower bounds on the residue at of this Dedekind zeta function which depend on , the absolute value of the discriminant of and the behavior in of the rational primes . Now, let be a real abelian number field and let be any real zero of the zeta function of . We give an upper bound on the residue at of which depends on , and the behavior in of the rational primes . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields which depend on the behavior in of the rational primes . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

     

On the Jacobi group and the mapping class group of

The paper contains a proof that the mapping class group of the manifold is isomorphic to a central extension of the (full) Jacobi group by the group of 7-dimensional homotopy spheres. Using a presentation of the group and the -invariant of the homotopy spheres, we give a presentation of this mapping class group with generators and defining relations. We also compute the cohomology of the group and determine 2-cocycles that correspond to the mapping class group of .

     

Boundary Hölder and estimates for local solutions of the tangential Cauchy-Riemann equation

We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood of a point when is a generic -concave manifold of real codimension in , where . Our method is to first derive a homotopy formula for in when is the intersection of with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any -closed form on without shrinking. We obtain Hölder and estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage d'un point d'une sous-variété générique -concave de codimension quelconque de . Nous construisons une formule d'homotopie pour le sur , lorsque est l'intersection de et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme -fermée sur . Nous en déduisons des estimations et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur .

     

Profinite and pro- completions of Poincaré duality groups of dimension 3

We establish some sufficient conditions for the profinite and pro- completions of an abstract group of type (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type over the field for a fixed natural prime (resp. of finite cohomological -dimension, of finite Euler -characteristic).

We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime .

     

Finiteness theorems for positive definite -regular quadratic forms

An integral quadratic form of variables is said to be -regular if globally represents all quadratic forms of variables that are represented by the genus of . For any , it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of variables that are -regular. We also investigate similar finiteness results for almost -regular and spinor -regular quadratic forms. It is shown that for any , there are only finitely many equivalence classes of primitive positive definite spinor or almost -regular quadratic forms of variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

     

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 .

     

Generalized interpolation in with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for functions on the unit disc to problems concerning lifting onto of an operator that is defined on ( is an inner function) and commutes with the (compressed) shift . In particular, he showed that interpolants (i.e., such that ) having norm equal to exist, and that in certain cases such an is unique and can be expressed as a fraction with . In this paper, we study interpolants that are such fractions of functions and are bounded in norm by (assuming that , in which case they always exist). We parameterize the collection of all such pairs and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.