Showing papers in "Crelle's Journal in 1987"

Relative bilinear complexity and matrix multiplication.

Abstract: The significance of this notion lies, above all, in the key role of matrix multiplication for numerical linear algebra. Thus the following problems all have \"exponent' : Matrix inversion, LK-decomposition, evaluation of the determinant or of all coefficients of the characteristic polynomial and for k = C also Qß-decomposition and unitary transformation to Hessenberg form. (See Strassen [48], [49], BunchHopcroft [12], Schönhage [45], Baur-Strassen [3], Keller [32].) Apart from this, such diverse computational problems äs finding the transitive closure of a finite relation, parsing a context free language or Computing a generalized Fourier transform are reducible to matrix multiplication. (See the survey article of Paterson [43], and Beth [4].) Of course the above definition of is incomplete: We have not explained what we mean by an algorithm or by the complexity of a computational problem. Fortunately, there is no need to do so. We may use instead the notion of rank of a bilinear map,

Constant mean curvature tori in terms of elliptic functions.

Abstract: Based on a numerical approximation of such a solution, we could produce plots of one ff-torus. In these Computer generated pictures the curvature lines for the smaller principal curvature λ^ looked almost planar. We then decided to restrict ourselves to fftori with one family of planar curvature lines. This condition translates into a second partial differential equation which induces a Separation of variables in the sinh-Gordon equation. Therefore the overdetermined System can be solved explicitly in terms of elliptic functions. We obtain a classification of all ff-tori in E which have one family of planar curvature lines.

Shimuravarietäten und Gerben.

Abstract: Das Problem der Fortsetzbarkeit der Hasse-Weil-Zeta-Funktionen und allgemeiner der motivischen L-Funktionen ist nach wie vor ein zentrales Problem der Zahlentheorie. Es wird oft in zwei Probleme aufgeteilt, die getrennt zu lösen sind. Es ist erstens zu zeigen, daß jede motivische L-Funktion gleich einer automorphen L-Funktion ist, und zweitens, daß jede automorpheL-Funktion fortsetzbar ist. Beide Probleme sind in herzlich wenigen Fällen gelöst und dann nur dank der Bemühungen vieler Mathematiker über lange Zeit.

Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems.

Abstract: Soit N(λ) le nombre de zeros d'une solution de l'equation de Sturm-Liouville ponderee generale (p(x)y')'+(λr(x)-q(x))y=0 sur [0,b] ou 0

Galois module structure of abelian extensions.

Abstract: The connection between Stickelberger relations and the structure of rings of integers s Galois modules is made apparent in Hilbert's proof [H], Theorems 135 and 136, of the formet for ideal classes in the prime cyclotomic field 0(μ^). The connection is made through the resolvents (α|χ) of elements α generating normal integral bases for certain cyclic extensions L/Q of prime degree /, namely those contained in the cyclotomic fields Q( p) where p is a prime, p^l (mod/).

Iwasawa theory for the symmetric square of an elliptic curve.

Abstract: Up until the present time, most work in Iwasawa theory has dealt with either the cyclotomic theory or descent theory on abelian varieties. We began work on the material in this paper several years ago in an effort to formulate precise questions of Iwasawa theory for more general L-functions which are of arithmetic interest. It seemed to us that the first case to consider was the L-function attached to the Symmetrie square of the Täte module of an elliptie curve defined over Q. The ahn of the present paper is to present the rather fragmentary results we have obtained in this direction, äs well äs several precise conjectures. Throughout, we have only considered primes p such that the elliptie curve has good ordinary reduction at p — the case of all other primes remains shrouded in mystery at present. Finally, we wish to express our thanks to R. Greenberg, whose many suggestions over the last year have greatly helped us. Indeed, Greenberg has now gone a long way towards formulating precise conjectures of Iwasawa theory for the L-function attached to an arbitrary /-adic representation and a prime p which is ordinary for this /-adic representation.

Singular limits for the compressible Euler equation in an exterior domain

Abstract: Justification des phenomenes lies au passage des equations des fluides parfaits compressibles a celles des fluides parfaits incompressibles Limites singulieres de l'equation d'Euler des fluides compressibles dans un domaine exterieur

Hereditary orders, Gauss sums and supercuspidal representations of GLN.

Abstract: We prove here two fundamental structure theorems on the behaviour of an irreducible adiiiissible representation of the general linear group G = GLN (F), of a padic local field F, when it is restricted to a suitable open compact or compaet-modcentre subgroup. The main one concerns supercuspidal representations and their restrietions to maximal compact-mod-centre subgroups (i.e. normalisers of principal Orders in A~MN(F)). It answers affirmatively a question, which had long since hardened into a conjecture, first posed in [2]. There are a number of direct consequences. First, we get an explicit formula for sthe Godement-Jacquet local constant ß(is, s) in terms of the Gauss sums of [2]. This gives a relation between the \"analytic conductor\" of n (defined from the local constant) and an \"algebraic conductor\" (coming from the group theory). We can also give a direct local proof of CarayoFs exhaustion theorem [3] for supercuspidals in the case where the dimension N is prime, avoiding the global methods implicit in Carayol's use of the matching theorem of Bernstein-DeligneKazhdan-Vigneras. We do not give any complete results of this kind for compound N, but it seems reasonable to speculate that our theorem will be helpful when trying to decide whether or not all supercuspidals are induced from maximal compact-mod-centre subgroups.

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

Abstract: The two fundamental finiteness theorems in the arithmetic theory of elliptic curves are the Mordell-Weil theorem, which says that the group of rational points is finitely generated, and Siegel's theorem, which asserts that the set of integral points (on any affine subset) is finite. Serge Lang ([4], p. 140) has conjectured a quantitative relationship between these two results, namely that for a given number field, the number of integral points on a quasi-minimal model of an elliptic curve should be bounded solely in terms of the rank of the group of rational points. In this note we will prove Lang's conjecture for the class of elliptic curves having integral j-invariant. More precisely, we will prove the following result. (See corollary 7. 2.)

Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces

Abstract: Soit X=G/K un espace de Riemann symetrique de type non compacte a bord B=K/M, et soit D(X) l'algebre de tous les operateurs differentiels G-invariants sur X. On obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur X

A formula for the Cartier operator on plane algebraic curves

Abstract: The main purpose of this paper is to establish a formula for the Cartier operator on a plane algebraic curve in terms of a differential operator in polynomials in two variables. This formula is very useful for computations and is also of theoretical interest To illustrate the latter we make an analysis of the ideals defining canonical curves in characteristic 2 and also give an elementary proof of a theorem of Manin on the number, modulo p, of rational points of an algebraic curve defined over a finite field of characteristic p.

Harmonic maps which solve a free-boundary problem

Abstract: The study of harmonic mappings has seen considerable progress in recent years. Most of the existence and regularity results for boundary-value problems have concerned themselves with the Dirichlet problem, in which the mapping is required to agree with a given mapping on the boundary of its domain. In the present paper, we shall address a free-boundary problem, which is a natural extension of the Plateau boundary conditions and the free boundary conditions for surfaces.

A potential well theory for the wave equation with a nonlinear boundary condition.

Abstract: Soit D un sous-ensemble connexe ouvert borne de R n a frontiere de Lipschitz ∂D. Soit ∂D l'union de deux sous-varietes disjointes a (n-1) dimensions σ et Σ et leur confluence de Lipschitz. On considere le probleme aux valeurs limites et initiales: ∂ 2 u/∂t 2 =Δu dans D×(o,T), u(x,o)=U(x), ∂u(x,o)∂t=V(x) dans D, u(x,t)=o sur σ×(o,T), ∂u/∂n=f(u(x,t)) sur Σ×(o,T)

Point interactions in two dimensions: basic properties, approximations and applications to solid state physics

Abstract: On etudie rigoureusement l'operateur de Schrodinger formellement donne par H=−Δ−μδ(•−y) dans L 2 (R 2 ) et on montre comment l'approcher en termes d'hamiltoniens a courte portee transformes d'echelles. On considere des questions semblables pour l'operateur H=−Δ−Σ j=1 N μ j δ(•−y j ), N∈N. On etudie les proprietes spectrales du modele a un electron d'un cristal a 2 dimensions avec des interactions ponctuelles

Sur l'existence des fibrés vectoriels holomorphes sur les surfaces non-algébriques.

Abstract: Soit X une variete C-analytique compacte, connexe. La question laquelle nous nous interessons est de determiner quels sont les fibres vectoriels topologiques (complexes) qui admettent une structure de fibre vectoriel holomorphe. Pour les fibres L de rang un, la question est resolue l'aide de la suite exacte exponentielle, ou Taide du theoreme de Lefschetz sur les classes de type (l, 1): il faut imposer que la classe de Chern ci(L) appartienne au noyau de la fleche canonique

Real forms of smooth del Pezzo surfaces.

Abstract: Work on real forms is also classical: for cubic surfaces, beginning with Schl fli [13]; for quartic curves the best reference is Zeuthen [21] (see also [5]); while for a surface of intersection of two quadrics there is the well-known classification of pencils of quadrics (the list of references is so long I will not attempt it here). Early work on this problem used direct geometrical methods, though group theory has been used for some time in problems about del Pezzo surfaces over various non-algebraically closed fields, e.g. [11]. We refer the reader to [17] and the recent survey [12] for an account of work in this area.

The genus of PSl2 (q).

Abstract: In [3] we computed the genus of the linear fractional group PS/2(p), where p*z5 is a prime. That is, we determined the least integer g such that PSl2(p) could occur äs an effective group of orientation preserving symmetries of Sg9 the closed orientable surface of genus g. More generally, given any finite group G we define the genus of G to be the least integer g so that G acts effectively and orientably on Sg. There is no loss of generality if we require the actions to be conformal, that is analytic in some complex structure on Sg9 since the positive solution of the Nielsen Realization Problem by [4], [2] implies that if G acts topologically on Sg then it can also act conformally with respect to some complex structure.

A quantitative version of Hurwitz' theorem on the arithmetic-geometric inequality.

Abstract: by letting ii = xf^0 and writing the form (1.2) Fn(x1,...,xn) = x; + . . .+xn -nx1-xn p s a sum of terms, each of which is non-negative. (Actually, Hurwitz worked with — , n but the difference is cosmetic.) His expression gives the form F2d s a sum of squares of forms. Hurwitz' construction was explicit, but it used a large number («d2) of squares. We give an alternate, algorithmic, construction which requires fewer (&6d) squares; it is based on a reduction used by Hurwitz and applies to a somewhat wider class of forms. Hurwitz makes no mention of the number of squares, but Hardy, Littlewood and Polya [3], p. 55, in presenting his method, note that F6 is a sum of 19 squares. (Had they used their own observation (p. 56) that every positive semidefinite binary form is a sum of two squares, they could have replaced \"19\" by \"13\".)

Bifurcation points and eigenvalues of inequalities of reaction-diffusion type

Abstract: On demontre certaines assertions concernant les points de bifurcation et les valeurs propres de systemes de reaction-diffusion avec des conditions unilaterales

Normal integral bases and complex conjugation.

Abstract: We consider the problem of determining, for a given family of Galois extensions N l K with Galois group A9 their relative Galois module structure, that is, the ring of integers ON s module over the group ring OKA. In order to get a grip on this problem we ask and answer here in a number of cases first the possibly more restricted question: what, if anything, obstructs the Galois modules from being free? The answer is: complex conjugation.

Variational inequalities and harmonic mappings.

Abstract: The problem we deal with in this paper, is to prove the regularity of energy minimizing maps. In this context the constraint u(x) e B for Jf \"-almost all χ € Ω plays the part of a geometric obstruction and we must face the difficulty that, generally, the minimizers will locally be in contact with the geometric obstacle dB and hence the minimum property will yield only a variational inequality instead of a variational equality for vectorfunctions.

K... of the cone over a curve.

Abstract: Let XaA + i be the affine cone over a smooth curve Cap> over an algebraically closed field of characteristic φ 2. In this note, we construct a homomorphism Φ from SK^Jf) to the non-zero finite dimensional fc-vector space H°(C, ω€(1)) (this vector space vanishes only if C is a line, so that X ^ Xu). If C is embedded via a complete linear system of degree d^2g + l, where g is the genus of C, then we construct certain classes in SK^JQ whose images under Φ span a non-zero /c-subspace of H°(C, coc(l)). In particular, we prove (char/c=i=2)

Regularity results for minimizing currents with a free boundary.

Abstract: The regularity question for minimal surfaces or minimizing integer multiplicity rectifiable currents in the context of geometric measure theory is naturally divided into three cases. In the classical Plateau problem where one is looking for a minimal surface spanning a given boundary one encounters the problem of interior regularity äs well äs the problem of regularity near the fixed boundary. If one considers the problem to minimize area among currents whose boundary or part of it is supposed to lie in a given hypersurface one in addition has to treat the problem of regularity near the free (part of the) boundary.