Showing papers in "Crelle's Journal in 1997"

F-modules: Applications to local cohomology and D-modules in characteristic p > 0

Abstract: Let F: Ä-mod -* Ä-mod be the Frobenius functor of Peskine-Szpiro [PSz], 1.1.2. An F-module, or, more accurately, an FÄ-module is a pair (Jt, 0), where M is an Ä-module and : M » F ( J f ) is an -module isomorphism. F-modules form an abelian category (Definition 1.1). It should be noted that our notion of F-module is, in a sense, dual to the notion of level (R, F)-module of Hartshorne-Speiser [HaSp], Sec.l. It has been known that local cohomology modules of R with support in any ideal / c R have the property that they are isomorphic to their own images under the Frobenius functor and this fact has been used, for example, by Hartshorne-Speiser [HaSp], Huneke-Sharp [HuSh], Peskine-Szpiro [PSz] and Sharp [Sh] to study local cohomology, but a systematic theory of modules having this property has not been constructed. One of our main goals in this paper is to develop such a theory.

On the second boundary value problem for equations of Monge-Ampère type.

Abstract: We prove the existence of globally smooth convex Solutions of MongeAmpere equations of the form detDw=/(x,M,Z)w) in Ω subject to the boundary condition Du(Q) = Ω* where Ω and Ω* are smooth uniformly convex domains in R\".

Normal polytopes, triangulations, and Koszul algebras.

Abstract: This paper is devoted to the algebraic and combinatorial properties of polytopal semigroup rings defined äs follows. Let P be a lattice polytope in IR, i.e. a polytope whose vertices have integral coordinates, and K a field. Then one considers the embedding : R\" -> R + , ( ) = (x,l), and defines SP to be the semigroup generated by the lattice points in i(P); the Ä^algebra K[Sp] is called a polytopal semigroup ring. Such a ring can be characterized äs an affine semigroup ring that is generated by its degree l elements and coincides with its normalization in degree l.

Stratification of minimal surfaces, mean curvature flows, and harmonic maps.

Abstract: In this paper we prove some abstract Stratification theorems for closed sets or more generally for upper semicontinuous functions. (Ciosed sets correspond to 0-1 valued upper semicontinuous functions.) The theorems can be applied to minimal varieties, mean curvature flows, and energy minimizing p-harmonic maps to give stratifications by tangent cone or blow-up type. For example, s special cases of the theorems proved here we have:

Amenability for Fell bundles.

Abstract: Given a Fell bundle B, over a discrete group Γ, we construct its reduced cross sectional algebra C∗ r (B), in analogy with the reduced crossed products defined for C∗-dynamical systems. When the reduced and full cross sectional algebras of B are isomorphic, we say that the bundle is amenable. We then formulate an approximation property which we prove to be a sufficient condition for amenability. A theory of Γ-graded C∗-algebras possessing a conditional expectation is developed, with an eye on the Fell bundle that one naturally associates to the grading. We show, for instance, that all such algebras are isomorphic to C∗ r (B), when the bundle is amenable. We also study induced ideals in graded C∗-algebras and obtain a generalization of results of Strǎtilǎ and Voiculescu on AF-algebras, and of Nica on quasi-lattice ordered groups. A brief comment is made on the relevance, to our theory, of a certain open problem in the theory of exact C∗-algebras. An application is given to the case of an Fn–grading of the Cuntz– Krieger algebras OA, recently discovered by Quigg and Raeburn. Specifically, we show that the Cuntz–Krieger bundle satisfies the approximation property, and hence is amenable, for all matrices A with entries in {0, 1}, even if A does not satisfy the well known property (I) studied by Cuntz and Krieger in their paper.

On Galois representations associated to Hilbert modular forms.

Abstract: In this paper, we prove that, to any Hilbert cuspidal eigenform, one may attach a compatible system of Galois representations. This result extends the analogous results of Deligne and Deligne–Serre for elliptic modular forms. The principal work on this conjecture was carried out by Carayol and Taylor, but their results left one case remaining, which we complete in this paper. We also investigate the compatibility of our results with the local Langlands correspondence, and prove that whenever the local component of the automorphic representation is not special, then the results coincide.

Bernoulli's free-boundary problem, qualitative theory and numerical approximation.

Abstract: Bernoulli's free-boundary problem arises in ideal fluid dynamics, optimal insulation and electro chemistry. In electrostatic terms we design an annular condenser with a prescribed and an unknown boundary component such that the electrostatic field is constant in magnitude along the free boundary. Typically the interior Bernoulli problem has two Solutions, an elliptic one close to the fixed boundary and a hyperbolic one far from it. Previous results mainly deal with elliptic Solutions exploiting their monotonicity s discovered by A. Beurling. Hyperbolic Solutions are more delicate for analysis and numerical approximation. Nevertheless we derive a second order trial free-boundary method, the implicit Neumann scheme, with equally good performance for both types of Solutions. Super linear convergence of a semi discrete variant is proved under a natural non-degeneracy condition. Numerical examples computed by this method confirm analytic predictions including questions of uniqueness, connectedness, elliptic and hyperbolic limits. 1. Interior and exterior Bernoulli problem The interior Bernoulli problem is the following. Given a connected domain in (T and a constant Q > 0, find a subset A c Ω and a potential u: Ω\Α -* i such that -Δι/ = 0 in Ω\Α, u = Ο οη ΘΩ, u = l on dA , du „ Λ . — = on dA. ov The potential u lives on the domain Ω\Α, typically an annulus (Fig. 1). The exterior unit normal of this domain is denoted by v. In the classical setting the free-boundary condition means 166 (1) Flucher and Rumpf , Bernoulli's free-boundary problem

Spectral analysis of long wavelength periodic waves and applications.

Abstract: This paper presents a spectral analysis of large wavelength periodic travelling wave Solutions of nonlinear evolutionary p.d.e. 's in one space variable. The framework is general enough to include waves occurring in a variety of different equations such äs the generalized KdV and other dispersive equations, and also, parabolic Systems. It is assumed that the equations admit a family of large wavelength periodic waves which, äs a parameter tends to zero, tend to a limiting homoclinic (or solitary) wave. In regions of the spectral plane in which the homoclinic wave has isolated eigenvalues, the main result, Theorem l .2, is that the periodic waves have continua of eigenvalues in a neighborhood of each isolated eigenvalue of the homoclinic wave for suffieiently small a. Generically, these continua will form loops in the spectral plane. The main result is applied to the spectral analysis of long wavelength periodic waves arising in several applications, including periodic viscous profiles of Solutions of degenerate 2 x 2 conservation laws, the generalized KdV, BBM, and Boussinesq equations, the FitzHugh-Nagumo System, and the Gray-Scott model.

An equivariant Riemann-Roch theorem for complete, simplicial toric varieties.

Abstract: Introduction. The theory of toric varieties establishes a now classical connection between algebraic geometry and convex polytopes. In particular, äs observed by Danilov in the seventies, finding a closed formula for the Todd class of complete toric varieties would have important consequences for enumeration of lattice points in convex lattice polytopes. Since then, a number of such formulas have been proposed; see [M], [Pl], [P2] The Todd class of complete simplicial toric varieties is computed in [G-G-K], using the Riemann-Roch formula of T. Kawasaki [Ka].

P-orderings and polynomial functions on arbitrary subsets of Dedekind rings.

Abstract: We introduce the notion of a P-ordering of an arbitrary subset AOf a Dedekind ring R, and use it to investigate the functions from X to R which can be represented by polynomials. In the case when R is a finite principal ideal ring, our results include canonical representations for those polynomials which vanish on X, a canonical representation for each polynomial function from X to , necessary and sufficient congruence conditions for a function from X to R to be a polynomial function, and a formula for the number of such functions. When R is a Dedekind domain, we use the concept of P-ordering to give necessary and sufficient conditions for the existence of a regulär basis for the ring lnt(X,R), and moreover, we give an explicit construction of such a basis whenever it exists. Finally, we deduce many of the classical theorems of Kempner, Carlitz, Polya, and others on polynomial mappings äs special cases of our results.

On new isoperimetric inequalities and symmetrization.

Abstract: Using recent isoperimetric inequalities for quermassintegrals on domains that are not necessarily convex, we develop a theory of symmetrization, extending the well known Schwartz spherical symmetrization. As applications, we derive sharp estimates for Hessian equations and Hessian integrals.

L-functions and periods of polarized regular motives.

Abstract: It is probably fair to say that all known results on special values of L-functions of motives over number fields are proved by identifying the L-function in question, more or less explicitly, with an automorphic L-function, and then applying analytic methods, derived in all but the simplest cases from the theory of automorphic forms. (Here and throughout the paper, the term \"motive\" is used äs in Deligne's article [D]: i.e., to designate a pure motive for absolute Hodge cycles.) It is unlikely that every motivic Lfunction can be handled in this way. The pupose of this paper is to call attention to a certain class of motivic L-functions that, conjecturally at least, can all be realized äs Lfunctions of cohomological automorphic forms, in most cases on Shimura varieties attached to unitary groups. These motives, which we view initially äs motives over Q, can be characterized by their Hodge types: they are regulär, in the sense that no Hodge type has dimension greater than l over a suitable coefficient field, and polarized in the usual sense. In general only the base change of such a motive M to an imaginary quadratic field Jf can be realized in the cohomology of unitary Shimura varieties; in compensation, one then has the freedom to twist the base change M# by a motivic Hecke character of Jf call the result M# ® without leaving the class of automorphic forms.

Some results on Q(p) spaces, 0lpl1

Abstract: For p is an element of (0, 1), let Q(p) be a proper subspace of BMOA defined by means of a modified Garcia norm. We give a boundary value criterion for a function to be in Q(p) and a necessary and sufficient condition in terms of p-Carleson measure for an

The number of curves of genus two with elliptic differentials.

Abstract: Let C be a curve of genus 2 defined over an algebraically closed field K, and suppose that C admits a non-constant morphism f : C → E to an elliptic curve E. If f does not factor over an isogeny of E, then we say that f is an elliptic subcover of C. Note that this last condition imposes no essential restriction since every nonconstant f : C → E factors over a unique elliptic subcover fmin : C → Emin. A classical theorem due to Picard [Pi] and Bolza [Bo] of 1882/86 states that a curve C of genus 2 has either none, two or infinitely many elliptic subcovers. This is in part due to the fact that the elliptic subcovers occur in pairs. More precisely, given an elliptic subcover f : C → E, there is a canonical “complementary” elliptic subcover f ′ : C → E ′ of the same degree N := deg(f) = deg(f ′) which is characterized by the requirement that the induced maps on the associated Jacobian varieties fit into an exact sequence

Uniruled projective manifolds with irreducible reductive G-structures

Abstract: We will call a G-structure modeled after a compact irreducible Hermitian Symmetrie space S of rank ^ 2, an S-structure. (See section 3 or [KO] for a precise definition. Note that our S-structure is called G(S)-structure in [KO].) Such structures were studied by many authors in the 60's (see [Oc] and the references there). From the 80's, they were studied by people working on twistor theory (see [Ba], [Ma] and the references there). When one studies these works, what is rather amazing, at least to the authors, is the lack of a nonflat example among compact manifolds. One may even expect that 5-structures are always flat under mild conditions. One result along this line is

Periods, poles of L-functions and symplectic-orthogonal theta lifts.

Abstract: As an application of their regularized Siegel-Weil formula, Kudla and Rallis prove in [K.R.], Theorem 7.2.6, that for an irreducible, automorphic, cuspidal representation π of Sp2n(A), the only possible poles of the Standard (partial) L-function L(n,s) are simple and occur at the points < l, 2,.. . , -l· l Λ If L(n,s) has a pole at a given point, then the theta lift of π, to an appropriate orthogonal group, is nonzero. The orthogonal group corresponds to a quadratic form, whose discriminant character is trivial. For example, at s = l, the quadratic form is in 2/7 variables. In this paper, we restrict attention to generic π only, i.e. those with nontrivial Whittaker coefficients, and then we show that • the only possible pole of L(n,s) is at s = l,

Markov traces and knot invariants related to Iwahori-Hecke algebras of type B.

Abstract: In classical knot theory we study knots inside the 3-sphere modulo isotopy. Using the Alexander and Markov theorem, we can translate this into a purely algebraic setting in terms of Artin braid groups modulo an equivalence relation generated by ‘Markov moves’ (one of which is usual conjugation inside the braid group). V.F.R. Jones [5] used this fact in 1984 for constructing a new knot invariant through trace functions on the associated Iwahori-Hecke algebras of type A with suitable properties that reflect the above Markov moves. Jones’s work led to questions of developing knot theory corresponding to other types of Coxeter groups. It is proved in [6] that there exist braid structures related to arbitrary 3-manifolds, which in addition satisfy appropriate Markovisotopy equivalence; also that, if the 3-manifold is a solid torus, then the sets of related braids form groups, which are in fact the Artin-Tits braid groups related to the B-type Coxeter groups. These results together with a linear trace that we found in 1991 are used in [7] for constructing a 4-variable analogue of the homfly-pt (2-variable Jones) polynomial for oriented knots inside a solid torus. We proved the existence of this trace (see [7]) by following and adapting to the B-type case Jones’s proof of the existence of Ocneanu’s trace in [5], Theorem 5.1.

Systole et invariant d'Hermite.

Abstract: Nous nous interessons ici certaines generalisations de l'invariant d'Hermite etudiees recemment par divers auteurs: invariant d'Hermite des varietes abeliennes principalement polarisees (P. Buser et P. Sarnak [B-S]), i.e. des reseaux symplectiques dont l'etude a ete developpee dans [B-M3], invariant d'Hermite restreint des familles de reseaux, par exemple les reseaux isoduaux orthogonaux et symplectiques (A.-M. Berge et J. Martinet [B-M3]), fc-invariant d'Hermite (R. Coulangeon [Cou]). Nous considerons egalement une generalisation un autre type d'objet geometrique: la systole des surfaces de Riemann munies de la metrique de Poincare (P. Schmutz [Schi]).

Tight closure and the Kodaira Vanishing Theorem

Abstract: In its familiär form, the Kodaira Vanishing Theorem is a Statement about the cohomology of the sheaf of sections of a positively curved line bündle <£ on a complex projective manifold X\\ it asserts that H(X9&~) = Q for / less than the dimension of X. However, the Kodaira Vanishing Theorem also has a purely ideal-theoretic Interpretation concerning primary decompositions of ideals generated by part of a System of parameters for the section ring determined by <£. Moreover, the Kodaira Vanishing Theorem has a tight closure formulation, tying this classic and central result in complex algebraic geometry to new advances in characteristic p commutative algebra.

Projective resolutions of representations of GL (n).

Abstract: The representations of the algebraic group GL(n) are well understood over a field of characteristic 0, but they are much more complicated in characteristic p (or over the integers), essentially because there can be nontrivial extensions between two representations. Akin-Buchsbaum [2] and Donkin [10] improved matters by showing that the category P (n, r) of polynomial GL(F)-modules (n = dim V) which are homogeneous of fixed degree r, such äs ® , AV9 S\"T, the irreducible subquotients of these modules, and so on, has finite global dimension. That is, there is an integer N depending on r and n = dim V such that every polynomial GL(F)-module of degree r has a resolution of length ^N by projective objects in the category of degree-r representations of CL(F). An equivalent Statement is that for polynomial representations A, B of degree r,

Subtriviality of continuous fields of nuclear C*-algebras, with an appendix by E. Kirchberg

Abstract: We extend in this paper the characterisation of a separable nuclear C* -algebra given by Kirchberg proving that given a unital separable continuous eld of nuclear C* -algebras A over a compact metrizable space X, the C(X)-algebra A is isomorphic to a unital C(X)-subalgebra of the trivial continuous fi eld O_2\otimes C(X), image of O2\otimes C(X) by a norm one projection.

On deformation of maximally degenerate stable marked curves and Oda's problem.

Abstract: We shall start with a construction involving an explicit parametrization of a universal deformation of such a degenerate curve X° (see § 1.2, §2). Then we shall study a certain 1-parameter subfamily of this deformation from the Galois theoretic viewpoint. Our first step for this is to construct a \"tangential base point\" on the total space of deformation, outside X° but near each of its singular points. Our explicit parametrization of deformation is crucial in this construction. Paths connecting these base points will then be compared with paths around and paths inside X°. (This, of course, involves comparison of Galois actions.) They are all \"within\" the formal neighborhood of X°. Since the family is 1-dimensional so that X° is a divisor, Grothendieck-Murre theory [GM] can be fully used. This study is presented in §3 (the main result is quoted in § 1.3 below).

Auto-intersection du dualisant relatif des courbes modulaires Xo (N).

Abstract: Une surface arithmetique est l'unique modele regulier minimal d'une courbe lisse et geometriquement connexe de genre g non nul sur un corps de nombres. Arakelov [1] a developpe en 1972 une theorie des intersections pour les diviseurs "compactifies" sur une teile surface. Cette theorie permet d'attacher ä la courbe des invariants numeriques analogues ä ceux definis dans le cadre geometrique: l'auto-intersection du dualisant relatif et le degre de son image directe. Ces invariants sont centraux dans les preuves de la conjecture de Mordell sur les corps de fonctions en toute caracteristique dues ä Parshin, Arakelov et Szpiro [21]. On s'interesse ici ä l'auto-intersection du dualisant relatif.

On Gel'fand-Graev characters of reductive groups with disconnected centre.

Abstract: Let G be a connected reductive algebraic group over an algebraic closure JFq of the finite field fq9 q being a power of p which is assumed throughout this work to be a good prime for G. Let F be a Frobenius endomorphism corresponding to some F€-structure on G. The purpose of this paper is, following the theme of [DLM], to present explicit results on the character theory of the group of rational points G, particularly in the case when the centre of G is not connected. We refer to [DLM] for an explanation äs to how the issues treated here relate to the determination of character values. Here we prove two results.