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).

     

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.

     

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.

     

Examples for the mod motivic cohomology of classifying spaces

Let be the classifying space of a compact Lie group . Some examples of computations of the motivic cohomology are given, by comparing with , and .

     

Connections with prescribed first Pontrjagin form

Let be a principal bundle over a manifold of dimension . If , then we prove that every differential 4-form representing the first Pontrjagin class of is the Pontrjagin form of some connection on .     

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.

     

Cyclicity of CM elliptic curves modulo

Let be an elliptic curve defined over and with complex multiplication. For a prime of good reduction, let be the reduction of modulo We find the density of the primes for which is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.

     

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.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

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 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.

     

Fitting's Lemma for -graded modules

Let be a map of free modules over a commutative ring . Fitting's Lemma shows that the ``Fitting ideal,' the ideal of minors of , annihilates the cokernel of and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a -graded skew-commutative algebra and prove corresponding theorems about the annihilator; for example, the Fitting ideal and the annihilator of the cokernel are equal in the generic case. Our results generalize the classical Fitting Lemma in the commutative case and extend a key result of Green (1999) in the exterior algebra case. They depend on the Berele-Regev theory of representations of general linear Lie superalgebras. In the purely even and purely odd cases we also offer a standard basis approach to the module when is a generic matrix.

     

Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector

Let be a -step nilpotent Lie algebra; we say is non-integrable if, for a generic pair of points , the isotropy algebras do not commute: . Theorem: If is a simply-connected -step nilpotent Lie group, is non-integrable, is a cocompact subgroup, and is a left-invariant Riemannian metric, then the geodesic flow of on is neither Liouville nor non-commutatively integrable with first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.

     

On the Diophantine equation : Higher-order recurrences

Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation

with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation

has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

     

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.

     

-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.

     

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.

     

Construction of -structures and equivalences of derived categories

We associate a -structure to a family of objects in , the derived category of a Grothendieck category . Using general results on -structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Belinson's equivalences.

     

Poset block equivalence of integral matrices

Given square matrices and with a poset-indexed block structure (for which an block is zero unless ), when are there invertible matrices and with this required-zero-block structure such that ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain . As one application, when is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of and have determinant . The invariants involve an associated diagram (the ``-web') of -module homomorphisms. The study is motivated by applications to symbolic dynamics and -algebras.