Showing papers in "Crelle's Journal in 1989"

Spectral curves and the generalised theta divisor.

Abstract: Let X be a smooth, irreducible, projective curve over an algebraically closed field of characteristic 0 and let g = gx ̂ 2 be its genus. We show in this paper that a generic vector b ndle on X of rank n and any fixed degree can be obtained s the direct image of a line b ndle on an n-sheeted (ramified) covering of X. From this we deduce results concerning linear Systems on the moduli space of vector bundles of rank n and degree 0.

More than two fifths of the zeros of the Riemann zeta function are on the critical line.

Abstract: In this paper we show that at least 2/5 of the zeros of the Riemann zeta-function are simple and on the critical line. Our method is a refinement of the method Levinson [11] used when he showed that at least 1/3 of the zeros are on the critical line (and are simple, äs observed by Heath-Brown [10] and, independently, by Seiberg). The main new element here is the use of a mollifier of length y=T with = 4/7 — whereas in Levinson's theorem the mollifier has 0 = 1/2 — . The work [6] of Deshouillers and Iwaniec on averages of Kloosterman sums is what allows us to use a longer mollifier. In fact, in their paper [7], they obtain an upper bound of essentially the right magnitude for an integral of the modulus squared of the zeta function multiplied by a mollifier of length T.

Real hypersurfaces with constant principal curvatures in complex hyperbolic space.

Abstract: Since E. Cartan's work in the late 30's the classification problem of hypersurfaces with constant principal curvatures is known to be far from trivial. In real space forms it leads to the well-known classification problem of isoparametric hypersurfaces, which has been solved in euclidean space by T. Levi-Civita [6] and B. Segre [11] and in hyperbolic space by E. Cartan [1]; in the sphere, however, a complete classification has not been obtained until now (for essential results see [1], [2], [3], [8], [9], [10], and the literature cited there).

On the total coloring of planar graphs.

Abstract: The total coloring of a graph G is a coloring of its vertices and edges in which any two adjacent or incident elements of F(G)u£(G) are colored with different colors. Behzad [1] and Vizing [9] conjectured independently that the total chromatic number χ, (G) of any graph G with the maximum degree A (G) is at most A (G) + 2. This conjecture was confirmed for A (G) = 3 by Vijayaditya [8] and for A (G) ̂ 5 by Kostochka [4]. The entire coloring of a planar graph G is a coloring of its vertices, edges and faces in a similar fashion. Kronk and Mitchem [6] conjectured the bound Xe(G) ^ A (G) + 4 and proved it for A (G) = 3. I proved in [2] :

Uniform p-adic cell decomposition and local zeta functions.

Abstract: The purpose of this paper is to give a cell decomposition for p-adic fields, uniform in p. This generalizes a cell decomposition for fixed p, proved by Denef [7], [9]. We also give some applications of our cell decomposition. A first implication is a uniform quantifier elimination for p-adic fields. Beiair [2], Delon [6] and Weispfenning [16] obtained quantifier elimination in other languages, but the language we use seems more practical for the evaluation of p-adic integrals. As a second application, we reprove results of Denef [10] on the dependence on p of the Igusa local zeta function. In this context we also obtain new results on p-adic integrals over sets definable in a language with cross section.

Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques.

Abstract: 0. Introduction. Soient pl5..., pr des points suffisamment generaux (ou generiques) du plan projectif et ml9 ..., mr des entiers positifs. Quelle est la dimension du Systeme lineaire des courbes planes de degre d passant par chaque pf avec la multiplicite m,? Dans [aHl], on a etudie certains aspects de cette question, faisant apparaitre des exceptions qu'on a eu la faiblesse de ne pas chercher ä comprendre. La presente note comble cette lacune, en eclairant ces exceptions par une conjecture, qu'on espere plus convaincante que celle avancee dans [aHl]. La nouvelle conjecture est naturelle des qu'on traduit le probleme initial en termes de diviseurs sur une surface rationnelle generique Sr: eile dit, grosso modo, qu'un diviseur sur Sr dont la trace sur toute courbe exceptionnelle de Sr est positive doit avoir la cohomologie naturelle, autrement dit doit etre non special (voir la definition plus bas). Cette conjecture est une Variante, qu'on espere plus expressive, d'une conjecture proposee par Harbourne ([bH2], 1.6.2). On explique ci-dessous comment cette conjecture permettrait de calculer la cohomologie de tous les diviseurs sur les surfaces rationnelles generiques et on recense les quelques cas ou la conjecture est demontree.

Bloc et séries de Lusztig dans un groupe réductif fini.

Abstract: De nombreux auteurs, ä la suite des travaux de Fong et Srinivasan sur les groupes lineaires et unitaires (cf. [Fo-Sr]), se sont attaques au probleme de la classification des /blocs des groupes reductifs finis sur un corps de cardinal q premier ä /: voir les memoires de Hiss [Hi] et de Schewe [Sc] sur les groupes exceptionnels, ou encore les travaux en cours de Fong et Srinivasan sur les groupes classiques.

The blocks of finite classical groups.

Abstract: In this paper we study the blocks of a finite classical group for primes r unequal to the defining characteristic p of the group, where r and p are odd primes. The relevant finite classical groups are groups of fixed points G of Frobenius endomorphisms F of classical algebraic groups G with connected centers. So G is a conformal symplectic or orthogonal group in even dimension, or a special orthogonal group in odd dimension, that is, one of the groups CSp(2w, q\\ CSO(2n,q\\ or SO(2n 4-1, #). The irreducible characters of these groups have been classified by Lusztig in [10]. In particular, his results imply a Jordan decomposition of the irreducible characters by pairs (5, /l), where s is a representative of a semisimple conjugacy class of the dual group G*, and is a unipotent character of (CG*(s))*. Our results imply a similar Jordan decomposition for blocks. Indeed, let us define a unipotent block to be a block containing a unipotent character. Then the blocks of G are labeled by pairs (s, ), where s is a representative of a semisimple r'-conjugacy class of G*, and : is a unipotent block of (CG*(s))*. These results are thus similar to those for blocks of general linear groups and unitary groups studied in [8], where the language of partitions, hooks, and cores allowed a combinatorial formulation for the classification of characters into blocks. In the present paper, a similar combinatorial formulation for the classification of characters into blocks is given äs well.

The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix.

Abstract: On considere des varietes M isometriques au quotient de l'espace hyperbolique a n dimensions H n par un groupe discret geometriquement fini T sans elements paraboliques et sans elements d'ordre fini. On demontre que la serie de Eisenstein associee admet un prolongement mesomorphe au plan complexe et on etudie l'operateur diffusion associe

The power of a prime that divides a generalized binomial coefficient.

Abstract: The main idea is to consider generalized binomial coefficients that are formed from an arbitrary sequence C, as shown in (3) below. We will isolate a property of the sequence C that guarantees the existence of a theorem like Kummer’s, relating divisibility by prime powers to carries in addition. A special case of the theorem we shall prove describes the prime power divisibility of Gauss’s generalized binomial coefficients [5, §5],

Some Groups of Transformations defined by Jordan Algebras. I

Abstract: It is well known that the Lie algebra of derivations of the exceptional Jordan algebra M 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional Lie algebra F 4 of Killing-Cartan and that the Lie algebra of linear transformations in M 3 8 leaving a certain cubic norm form defined on M 3 8 invariant is the exceptional Lie algebra E 6 (Chevalley-Schafer [7], Freudenthal [11]). The program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple Jordan algebras. The present paper gives the definition of the groups and Lie algebras in the general case and gives their determination for special Jordan algebras with some restrictions on the base field. In a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple Jordan algebras and in a third paper we hope to study the generalized groups of types E 6 which arise from these Jordan algebras.

Boundary estimates for solutions of nonlinear degenerate parabolic systems

Abstract: On etablit la regularite partout jusqu'a la frontiere pour des solutions faibles de systemes paraboliques du type ∂u (l) /∂t−div(|Du| p−2 Du (l) =f (l) (x,t,u,Du) dans Ω T ; l=1,2,…,m; u:Ω T →R m , m∈N, p>max{1;2N/(N+2)}. Ω est un ouvert borne de R N , N≥2, 0

Minimizing p-harmonic maps into spheres

Abstract: d m fif Qf where |V/| = £ Σ γ -~-^-, and where ya (x) = (/^(x))\" represents the metric i = l α,0 = 1 ^Χα νβ of M. In a fundamental paper ([SU 1]), Schoen and Uhlenbeck showed that near any singularity, a minimizing harmonic map /: M —·> N converges strongly to a minimizing tangent map u: M —»JV\", which is harmonic and homogeneous of degree zero. The investigation of minimizing tangent maps u: B —+ N is therefore an important aspect of current research into minimizing harmonic maps.

Diskontinuitätsbereich für arithmetische Äquivalenz

Abstract: Die vorliegende Arbeit benutzt mehrfach Methoden, welche von Dirichlet ausgebildet worden sind.

The cyclic homology and K-theory of curves

Abstract: Until recently, very little has been known about the higher algebraic X-theory of anything except for finite fields. The goal of this paper is to compute the X-theory of singular curves in characteristic zero, assuming the X-theory of smooth curves and fields äs given. We do this by first Computing the Hochschild and cyclic homology of the singular curves (using a new procedure which works more generally), and then invoking several recent results relating X-theory to the cyclic homology of 0-algebras, Q being the field of rational numbers.

The distribution of squarefree numbers.

Abstract: On etudie le comportement de la somme N(n,k)=Σ 0≤h

Courbes de Weil semi-stables de discriminant une puissance m-ième.

Abstract: Soit E une courbe elliptique semi-stable sur Q. Notons Δ son discriminant minimal. Soit m un entier ^1. Supposons que E soit une courbe de Weil (ce qui est conjecturalement toujours le cas) et que \\A\\ soit la puissance w-ieme d'un nombre entier. Nous demontrons en utilisant un theoreme de Ribet que Γόη a alors m ̂ 5 et que E possede un point d'ordre m rationnel sur Q (cf. theoreme 1). On en deduit qu'une courbe elliptique de Weil de conducteur premier p a en general pour discriminant ±p (la liste des exceptions est decrite dans le theoreme 2).

A trace formula for Jacobi forms.

Abstract: The purpose of this paper is to state and prove a general trace formula for Jacobi forms. The exact Statement of this trace formula can be found in § l, especially Theorem 1. In this Statement there occurs a certain quantity Gm(

A semi-linear elliptic equation in a strip arising in a two-dimensional flame propagation model.

Abstract: On montre l'existence d'une solution stationnaire pour un probleme a 2 dimensions pour l'equation elliptique semilineaire de la forme −Δu+v(y)u x =g(u) dans la bande S={(x,y)∈R 2 , x∈R, 0 0

On the asymptotic distribution of the eigenvalue branches of the Schrödinger operator H±λW in a spectral gap of H

Abstract: On considere des operateurs de Schrodinger H=−Δ+V dans l'espace de Hilbert L 2 (Rν) avec un potentiel V satisfaisant 1≤VeL∞(Rν). On suppose que l'intervalle (a,b) est contenu dans l'ensemble resolvant de H. On obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans (a,b) des familles d'operateurs H±λW, λ>0, quand λ→∞, ou W est un potentiel a courte portee non negatif

L-functions attached to Jacobi forms of degree n. Part I. The basic identity.

Abstract: Let F be a Siegel modular form of degree n; thus F is an automorphic form on Gn = Spn. If F is a cuspidal common eigenform under the Hecke operators, we can attach to F the Standard zeta function L (s, F) corresponding to the Standard representation of the L-group G° = SO(2n +l, C). Its analytic continuation and functional equation were first studied by Andrianov and Kalinin [1] and completely established by Böcherer [4] and, independently, by Piatetski-Shapiro and Rallis [17].

Some groups of transformations defined by Jordan algebras. II. Groups of type F4.

Abstract: The principal objective of this paper is the study of one of the types of linear groups defined in I [15]: the groups of automorphisms of central simple exceptional Jordan algebras. This can also be considered as the second step in a program of studying “exceptional” linear groups analogous to the exceptional simple Lie groups via non-associative algebras. The first step in this program has been taken in another paper ([13]) in which we studied the counterparts of the Lie groups of type G 2 as groups of automorphisms of Cayley algebras. In an analogous fashion the present paper deals with the counterparts of the Lie group F 4 as groups of automorphisms of exceptional Jordan algebras.

Surfaces with K2 = 2, pg = 1, q = 0

Abstract: In the classification of algebraic surfaces of general type, one of the most challenging problems is the complete description of the surfaces with small K. Also, surfaces with K — l92;pg = l,q = Q have provided counter-examples to both the local and global Torelli problems ([Cal], [Ca2], [Ch], [Toi], [Us2]). But, whereas the geometry of surfaces with X = l, pg = l, q = Q is completely understood ([Cal], [To2]), there is no systematic treatment of the case K = 2.

Connections between Morse sets for delay-differential equations.

Abstract: On etudie la dynamique globale de l'equation differentielle a retard scalaire x˙(t)=−f(x(t), x(t−1)) avec des conditions appropriees sur f

Harmonic analysis of the de Rham complex on the sphere.

Abstract: One may wonder why these results did not appear in the literature thirty years ago. Indeed, their purely representation-theoretic component (Theorem A below) is essentially well-known. The case of one-forms is worked out in Koränyi-Vägi [4]; see also Reimann [5]. However, the explicit identification of all the irreducible subspaces for forms of arbitrary degree and the computation of d, d*9 and on them has appearently not been done before.

Orders in quaternion algebras.

Abstract: On etudie certains ordres dans des algebres de quaternions et on developpe une formule de trace calculable pour les matrices de Brandt associees a ces ordres speciaux

On the existence of weak solutions of perturbated systems with critical growth

Abstract: Sur un domaine borne Ω de R N on considere le probleme de Dirichlet pour le systeme elliptique: A(u)+G(u)=f sur Ω et u/ ∂Ω =0, ou A(u) est un operateur quasi lineaire et la perturbation G(u) depend de u et de D(u). On etudie des conditions suffisantes sur A(u) et sur G(u) pour l'existence de solutions faibles

On the existence of curves with maximal rank in Pn.

Abstract: We work over an algebraically closed field of characteristic zero. Let ρ(ά, g, n):=(n -f 1) d — ng — n(n + 1) be the Brill-Noether number. From Brill-Noether theory i t is known that if ρ ̂ 0 there exists a unique irreducible component, P>(d, g, n), of H(d, g, n) (the Hubert scheme of smooth, connected curves of degree d, genus g in P) which has general moduli. This means that the functorial rational map from P(d, g, n) to the moduli scheme Mg of curves of genus g is dominant. Hence in the r nge ρ(ά, g, n)^0 we have a nice component of H(d, g, n). Then two problems arise naturally: