Showing papers in "Annals of Mathematics in 1973"

On Modular Forms of Half Integral Weight

Abstract: The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, but the forms of half integral weight have attracted only casual attention, in spite of their importance and ancientness. Indeed, the connection of such forms with zeta functions was never clarified. When Hecke developed his theory of Euler product forthe forms of integral weight, he pointed out the impossibility of a similar theory for the forms of half integral weight, and that only partial information could be obtained for the Fourier coefficients of such forms (Werke, p. 639). He explained this in more detail in his last paper [3], which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases. A treatment of a more general type of modular form was given by Wohlfahrt [12]. In fact, he defined Hecke operators whose degree is the square of a prime, and showed a certain multiplicative relation, as predicted by Hecke, for the Fourier coefficients, but discussed neither Euler product, nor connection with zeta functions. In the present paper, we try to reveal a more affirmative aspect of the subject. To be specific, put, for each positive integer N,

Resonances in n-Body Quantum Systems With Dilatation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory

Abstract: It is our goal in this paper to give a precise mathematical definition of the notion of "resonance" in a class of n-body non-relativistic quantum systems and to begin a systematic development of the theory of such resonances. While this class of n-body systems is rather small when viewed in relation to the class of systems [54] for which most of the standard quantum mechanical lore can be developed, it is large enough to include systems with two-body Coulomb, Yukawa or Yukawian interactions. The class thus includes the systems of greatest importance to physics and, in particular, it includes the standard non-relativistic model of the atom. The principal line of development in this paper and the major new results which we wish to prove concern the so-called "time-dependent perturbation theory", one of the two standard perturbation theories developed during the earliest days of quantum mechanics. The other standard theory, known as "time-independent" or Rayleigh-Schrodinger perturbation theory, has been on a firm mathematical footing since the work of Rellich [46]. (Important refinements of Rellich's theory are due to Kato [35] and Sz-Nagy [59].) The time-dependent theory on the other hand has resisted a general mathematical formulation for over forty years although there has been some partly successful work on the subject which we will review later in this introduction. To avoid the natural confusion between "time-dependent" and "time-independent" we will generally avoid the use of the latter term, employing "Rayleigh-Schrodinger" and "Kato-Rellich" instead. The lowest order terms in the time-dependent perturbation series were developed in the 1920's as a means of computing radiative lifetimes of excited states of atoms. The quantity which is supposed to be approximated by this series is the inverse of the lifetime, z, which was assumed to be

Classification of subshifts of finite type

Abstract: Introduction 120 1 Definitions 122 2 Statement of results 126 3 Markov partitions and maps of 1-sided shifts 128 4 Existence and uniqueness of 1-sided shifts 132 5 Subdivision matrices and the proof of Theorems H and I 137 6 Ladder maps and conjugacies 142 7 Shift equivalence = strong shift equivalence 144 8 Proof of Theorem B 148 9 Proof of Theorem C 148 10 Realization theorem 150 11 The examples 151 Bibliography 152

Iterated Integrals of Differential Forms and Loop Space Homology

Abstract: We continue to define iterated integrals 5w ... w, where wl, *., w, are 1forms. Such iterated integrals will be defined for forms wi, *-*, w, of arbitrary degrees on a differentiable manifold or, more generally, a suitably defined "differentiable space". The space P(X) of piecewise smooth paths on a differentiable manifold X turns out to be a differentiable space. If wi, *.., w, are forms of respective degrees P1, ..., Pr on X, then the iterated integral Wi .. Wr is a form of degree Pi + * -1+ Prr on P(X). Let P(X; x0, xl) denote the subspace of P(X) consisting of paths from a given point x0 to another given point xl. We are interested in those iterated integrals or linear combinations of iterated integrals, which give rise to de Rham cohomology classes of QSX = P(X; x0, x0) and shall prove an iterated integral version of a loop space de Rham theorem, which includes the following

Pointwise Convergence of Fourier Series

Abstract: In this paper, we present a new proof of a theorem of Carleson and Hunt: The Fourier series of an LP function on [0, 2J] converges almost everywhere (p > 1). (See [1], [51.) Our proof is very much in the spirit of the classical theorem of Kolmogoroff-Seliverstoff-Plessner [8]. Unlike Carleson's proof, which makes a careful analysis of the structure of an L2 function f, our arguments essentially ignore f, and concentrate instead on building up a basic "partial sum" operator from simpler pieces. Our methods are (almost) entirely L2. Sections 1-7 of this paper contain a proof of pointwise convergence for L2 functions; Section 8 contains the modifications necessary to handle LP, and includes various further comments.

Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin's braid group

Abstract: Let X, X be orientable surfaces. Let (p, X, X) be a regular covering space, possibly branched, with finitely many branch points and a finite group of covering transformations. We require also that every covering transformation leave the branch points fixed. A homeomorphism g : X -> X is said to be \"fiber-preserving\" with respect to the triplet (p,X,X) if for every pair of points x , x ' e l the condition p(x) = p(x') implies pg(x) = pg(x'). If g is fiber-preserving and isotopic to the identity map via an isotopy gs, then g is said to be \"fiber-isotopic to 1\" if, for every s e [0,1], the homeomorphism gs is fiber-preserving. The condition that an isotopy be a fiber-isotopy imposes a symmetry which one feels, intuitively, is very restrictive. However, we find

Local cohomological dimension of algebraic varieties

Abstract: If X is a smooth scheme of characteristic zero and Yc X is a closed subset, we find topological conditions on the singularities of Y which determine the best possible vanishing theorem for the sheaves of local cohomology Hf(F) for all i > r and all quasicoherent F. Applications include computation of the cohomological dimension of Pn - Y for arbitrary closed subsets Y and extensions of theorems of Lefschetz and Barth to the case of singular varieties.

On Local Solvability of Linear Partial Differential Equations

Abstract: The title indicates more or less what the talk is going to be about. I t is going to be about the problem which is probably the most primitive in partial differential equations theory, namely to know whether an equation does, or does not, have a solution. Even this is meant in the most primitive terms. I would like to begin by explaining what the terms are. As you all know, the really difficult analysis these days, and perhaps always, is the global analysis. Well, the problem that I am going to discuss is purely local—in the strictest possible sense: we would like to find out if a linear partial differential equation, with coefficients as smooth as you wish, admits locally a solution. Obviously, in this connection, negative results are very important: and negative results about local solvability have global implications. But of course positive results have also their importance. Let us state precisely what is the problem. The partial differential equation under study will be

Flag-Transitive Subgroups of Chevalley Groups

Abstract: Publisher Summary Let G be a finite group of Chevalley type and B the Borel subgroup of G. If L ≤ G and G = BL, then L is said to be flagtransitive. D. Higman determined all flag-transitive subgroups of G in case G is of type A n . This chapter presents an extension of Higman's theorem to the case of a general group of Chevalley type. The result is used to show the nonexistence of 2-transitive permutation representations of certain groups of Chevalley type.

On the Geometry of a Theorem of Riemann

Abstract: Let C be a smooth complete algebraic curve. Let I: C-+J be an universal abelian integral of C into its Jacobian J. Furthermore, let I(i): C(i) J be the mapping sending a point c1 + *-+ ci in the ith symmetric product C(i) to I(c1) + *-+ I(ci). Let Wi be the image of I(i) when i is less than the dimension of J. My purpose is to describe the tangent cone of Wi at any point. Let L(w) be the inverse image by I(i) of a point w in J. L(w) is a projective space representing a complete linear system of effective divisors on C. Geometrically, my main result is

The Surjectivity of Invariant Differential Operators on Symmetric Spaces I

Abstract: An hydraulic jack for an elevator installation comprises a pair of cylinders and a common plunger. One cylinder is fixedly mounted in a vertical position in a hoistway; while one end of the plunger is mounted within the fixed cylinder for vertical reciprocating movement therein. The other cylinder, which carries an elevator cab up and down in the hoistway, is slidably mounted about the other end of the plunger for vertical reciprocating movement thereon.

Normalized period matrices II

Abstract: curves in the first article (herafter denoted NPM I, Ann. of Math. 94 (1971), 337-388) of this series. As noted at that time the basic theme is the p-adic reducibility of Fuchs-Picard differential equations and the relation between this reducibility and the Newton polygon of the zeta function of the reduced hypersurface. These relations are not completely understood even in the case of curves where very little is known in the case of supersingularity. Our ignorance in the case of hypersurfaces is naturally more extensive. Roughly speaking the amount of information obtained depends upon the extent to which the Newton polygon (of the middle dimensional factor of the zeta function of the reduced hypersurfaces) approximate the "ideal" minimal form. The most extensive information exists for the case covered by Lemma 5.1 below. Except in ? 6 below, we restrict our attention to hypersurfaces of degree not divisible by p (cf. [2]). To indicate the nature of our investigation from another point of view we shall state a conjecture concerning curves which may be generalized in various ways and extended to varieties of higher dimension. Consider an algebraic system of plane curves defined over the prime field, Fp, and with affine N space as base space. For X in base space, let CR be the corresponding curve. We assume that for X generic the curve CR is non-singular

Patch Spaces: A Geometric Representation for Poincare Spaces

Abstract: Section Page Preface 306 ?1. Preliminaries 307 2. Elementary properties 310 ? 3. Surgery on patch spaces 311 ?4. Changing patch structures 323 ?5. Obstruction to transversality 329 ? 6. The existence and uniqueness of patch space structures on Poincare duality spaces 334 S 7. Applications 338 (7.0) Surgery composition formula (7.1) Surgery product theorem (7.2) Computing cobordism classes of Poincare spaces (7.3) Representing homology classes by Poincare spaces (7.4) Characteristic variety theorem and homotopy type of BSF (7.11) Patch space structures for spaces which are Poincare mod a set of primes

PL Characteristic Classes and Cobordism

Abstract: The Z2-cohomology of an iJ-space is a Hopf algebra over the Steenrod algebra. R. J. Milgram [5] has determined H*(G) and H*(BG), and D. Sullivan [7] has determined J7*(G/PL) and H*(Q(G/PL)). Our main results, determining the Hopf algebra structure of H*(PL) and H*(BPL), follow from spectral sequence arguments, once we have determined the map H*(G/PL)-*H*(G). W. Browder, A. Liulevicius, and F. P. Peterson [ l ] have shown that there is an isomorphism of rings 9l*~3l^®H*(BPL)//H*(BO), where 91* is the unoriented, differentiate cobordism ring determined by Thorn. Thus our homology computations are sufficient to determine 9d*. Our methods also determine if* (TOP) and H*(BTOP) as Hopf algebras. In fact, these computations are easier than the PL computations. The Kirby-Siebenmann topological transversality theorem implies that 9lï~7r*(ikTTOP) = Vl%®H*(BTOP)//H*(B0) in dimensions 7^4.