Showing papers in "Bulletin of the American Mathematical Society in 1989"

A surprising higher integrability property of mappings with positive determinant

Abstract: Introduction. Let Q be a bounded, open set in R\", n > 2, and assume that u: Q -+ R belongs to the Sobolev space W*(Çl\\R), i.e. IMIî*a« = fçi\\\\ + \\Du\\ dx < oo, where Du denotes the distributional derivative. Then det Du is, of course, integrable. The aim of this note is to show that under the additional assumption that detDw > 0 (almost everywhere) in fact detZ>wln(2 + det Du) is integrable (on compact subsets K of fi). When applied to a sequence of mappings u : Q —• R\" with det Du > 0, llw^H î,/! < C, this higher integrability result implies that the sequence detDu^ is weakly relatively compact in L(K). This allows us to improve known results on weak continuity of determinants [R, B] and existence of minimizers in nonlinear elasticity [BM]. In the terminology of Lions [L1,L2] and DiPerna and Majda [DM], the constraint det Du^ > 0 prevents the development of 'concentrations' in the sequence det Du^\\ One might ask whether analogous results hold for orientation preserving mappings between oriented compact Riemannian manifolds. In short, the function det Du ln(2 + det Du) is still integrable, but not necessarily uniformly so along a sequence which is bounded inW>. 'Concentrations' may occur, but only in a particular fashion (see [M]).

Knots are determined by their complements

Abstract: This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. The notion of equivalence of knots can be strengthened by saying that K and K' are isotopic if the above homeomorphism h is isotopic to the identity, or, equivalently, orientation-preserving. The analog of Theorem 1 holds in this setting too: if two knots have complements that are homeomorphic by an orientation-preserving homeomorphism, then they are isotopic. Theorem 1 and its orientation-preserving version are easy consequences of the following theorem concerning Dehn surgery.

The state of the second part of Hilbert's fifth problem

Abstract: \"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1,61, 389) with so much ingenuity...and other equations occurring in the literature of mathematics, do not directly involve anything which necessitates the requirement of the differentiability of the accompanying functions... In all these cases, then, the problem arises: In how far are the assertions which we can make in the case of differentiatie functions true under proper modifications without this assumption?\" (Hilbert's emphasis).

Galois representations for Hilbert modular forms

Abstract: Introduction. It is a basic problem of number theory to classify algebraic extensions of a number field F. For extensions with abelian Galois group, this is accomplished by class field theory. The main theorem of class field theory provides a canonical isomorphism between the Galois group Ga\\(F/F), where F is of the maximal extension of F with abelian Galois group, and the group no(Cf) of generalized ideal classes. Although the nonabelian case of this theory remains largely undeveloped, the conjectures of Langlands provide a framework. A key step in this approach is to dualize, thereby viewing the isomorphism of class field theory as a correspondence between the (continuous) complex, one-dimensional representations of Ga\\(F/F) and 7r0(Cy). Furthermore, L-series are attached to representations of both groups and Artin's reciprocity law asserts that these L-series coincide under the correspondence. The complex onedimensional representations of no(Cp) are of finite order and correspond to automorphic forms of a special type on GL\\, namely, to those whose infinity type is of finite order. Let A/r be the adele ring of F. The considerations above lead to a more general problem: for all n > 1, to identify the L-functions of automorphic forms on GLn(Af) of arithmetic type at infinity (cf. [BRn]) with L-functions attached to «-dimensional motivic Galois representations. By a motivic representation we mean one which occurs as a subrepresentation of the (étale) cohomology of a smooth proper variety defined over F. All complex Galois representations with finite image are motivic, as are certain A-adic representations with infinite image. Here we recall that a kadic representation is a continuous representation of Gd\\{F/F) on a finitedimensional vector space over a finite extension of Q/. Even for n = 1, this program goes beyond class field theory, because the one-dimensional motivic /Uadic representations may have infinite order (e.g., the representations provided by the Shimura-Taniyama theory of abelian varieties with complex multiplication). In this case, such a representation is Hodge-Tate [F] and hence, by a theorem of Tate, is locally algebraic. It is therefore associated to an algebraic Hecke character with the same L-function. Observe that our problem consists of two parts. On the one hand, given a Galois representation p, one wants to construct an associated automorphic form n{p). If n = 2 and p is a complex representation with solvable

Operator theory and algebraic geometry

Abstract: On presente des resultats en theorie des operateurs multivariable dont les demonstrations se relient a des techniques issues de la geometrie algebrique

A nonvanishing theorem for derivatives of automorphic $L$-functions with applications to elliptic curves

Abstract: 1. A brief history of nonvanishing theorems. The nonvanishing of a Dirichlet series 2 a(n)n~\\ or the existence of a pole, at a particular value of s often has applications to arithmetic. Euler gave the first example of this, showing that the infinitude of the primes follows from the pole of Ç(s) at s = 1. A deep refinement was given by Dirichlet, whose theorem on primes in an arithmetic progression depends in a fundamental way upon the nonvanishing of Dirichlet L-functions at s = 1. Among the many examples of arithmetically significant nonvanishing results in this century, one of the most important is still mostly conjectural. Let E be an elliptic curve defined over Q: the set of all solutions to an equationy = x-ax-b where a, b are rational numbers with 4a-27b ^ 0. Mordell showed that E(Q) may be given the structure of a finitely generated abelian group. The Birch-Swinnerton-Dyer Conjecture asserts that the rank of this group is equal to the order of vanishing of a certain Dirichlet series L(s,E) as s = 1—the center of the critical strip—and that the leading Taylor coefficient of this L-function at s = 1 is determined in an explicit way by the arithmetic of the elliptic curve. We refer to the excellent survey article of Goldfeld [5] for details. In 1977, Coates and Wiles [3] proved the first result towards the BirchSwinnerton-Dyer conjecture. The conjecture implies that if the L-series of E does not vanish at 1, then the group of rational points is finite. Coates and Wiles proved this last claim in the special case that E has complex multiplication (nontrivial endomorphisms). In this note, we announce a nonvanishing theorem which, together with work of Kolyvagin and Gross-Zagier, implies that E(Q) is finite when L(l,E) ^ 0 for any modular elliptic curve E. (A modular elliptic curve is one which may be parametrized by automorphic functions. Deuring proved that all elliptic curves with complex multiplication are modular; Taniyama and Weil have conjectured that indeed all elliptic curves defined over Q are modular.) Before giving details of our theorem, we mention several other nonvanishing theorems and arithmetic applications. The following discussion is necessarily not a complete survey Shimura showed that there is a correspondence between modular forms ƒ of even weight k and modular forms ƒ of half-integral weight (k + l)/2.

The nondegeneration of the Hodge-de Rham spectral sequence

Abstract: Using an integrable, homogeneous complex structure on the compact group SO(9), we show that the Hodge-de Rham spectral sequence for this non-Kahler compact complex manifold does not degenerate at Ei, contrary to a well-known conjecture. On any (compact) complex manifold M, the algebra of global complexvalued C°°-differential forms a*(M) has a bigrading given by the Hodge type; and the corresponding decomposition of the de Rham differential d = d + d gives rise to a double complex (a*'*(M),d, d). The spectral sequence corresponding to the first ("holomorphic") degree is E\ = H<*{M, QM) =» HPp(M) with dx = d : this is the Hodge-de Rham (HdR) spectral sequence. When M is compact and Kahler, E\ = Eoo by Hodge theory. A folklore conjecture of about thirty years' standing says that for any compact complex manifold one should have E2 = E^. We will give an example to show that this conjecture is false. 1. There is an old observation of H. Samelson (see Wang [4]) that every compact Lie group G of even dimension (equivalently even rank) can be made into a complex manifold in such a way that all left-translations by elements g e G are holomorphic maps: we call such a structure an LICS on G (= left invariant, integrable complex structure on G). If G is in addition semisimple, then no LICS can be Kahlerian because H(G : R) = 0. Our example is a particular LICS on SO(9) (equivalently Spin(9): see §3 below), for which E2 has complex dimension 26. Since one knows that E^ has complex dimension 16, we obtain the required nondegeneration of HdR. Received by the editors November 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32C10, 58A14, 55J05, 32M10. 1 Samelson's paper appeared in a Portuguese journal, which is perhaps understandably not available in Indian libraries. ©1989 American Mathematical Society 0273-0979/89 $1.00 + $.25 per page 19

Controlled topology in geometry

Abstract: The purpose of the present note is to announce some finiteness theorems for classes of Riemannian manifolds (cf. A, B and D below). Let ^*$(n) denote the class of closed Riemannian «-manifolds with sectional curvatures between k and K, diameter between d and D, and volume between v and V. Here k < K are arbitrary, 0 < d < D, and 0 < v < V, THEOREM A. For n ^ 3,4 the class ^£%'y{n) contains at most finitely many diffeomorphism types.

Nonlinear Fourier analysis

Abstract: On presente l'analyse de Fourier non lineaire sur un certain nombre d'exemples. On etudie les techniques generales. On considere certains aspects de la theorie de Littlewood-Paley

Progress in the theory of complex algebraic curves

Abstract: We describe the contemporary view of the theory of algebraic curves over the complex numbers, with emphasis on the moduli spaces of curves and linear series on them. We then give an exposition of some of the recent work on the question of the rationality of the moduli space. Introduction. In the last twenty years there has been a major development in our understanding of algebraic curves. A number of the classical problems have been solved and new directions of investigation have been begun. We will describe some of the history of the theory, and how it led to the modern point of view, and we will sketch proofs of many of the main assertions. Then we will explain something of how the modern ideas have been used to solve some old problems. Many features of the current wave of progress are closely connected with patterns that go back to the earliest period in the development of the theory, so it's best to start with ancient history. To talk of complex projective curves, you need the complex numbers and you need projective space, so \"ancient history\" for us will start when these things become available, between about 1800 (Gauss' proof of the fundamental theorem of algebra) and 1830, (the introduction by Plücker of homogeneous coordinates for the projective plane). It goes without saying that the history below is that of a Mathematician and not of an Historian—it should probably be described as \"fictionalized.\" Naturally we will have to leave out parts of the theory of curves at least as rich as the parts we can put in. We beg pardon in advance from anyone whose favorite bit we've skipped. A more detailed development along the lines of this article may be found in lectures from the Bowdoin conference (Harris [1988]). A very beautiful survey covering a different range of topics is that of Mumford [1976]. A curious aspect of the history of algebraic curves, as with other parts of algebraic geometry, is the breakdown of rigor which deeply affected the theory, leading to its virtual stagnation in a mire of unproved assertions and incomplete proofs in the end of the first third of this century. The late nineteenth and early twentieth centuries were of course a period of enormous vitality in algebraic geometry, a period in which a large fraction of our current knowledge of curves and surfaces was obtained. In some cases the low rigor was simply the result of the fact that people lacked a Received by the editors December 13, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14H10, 14-D2. Both authors are grateful to the NSF for partial support during the preparation of this

Besicovitch meets Wiener—Fourier expansions and fractal measures

Abstract: (3) v = fdju + vc for the decomposition into discrete (ƒ dju as above) and continuous vc parts. Let F = û = J2k' + J'e' dvc{y) be the Fourier transform of the measure v. Then Wiener's theorem says (2) continues to hold. This means that the Fourier transform vc of the continuous portion of the measure does not contribute to the Bohr mean of |F|. We will interpret Wiener's theorem to mean that every finite measure is in some sense zerodimensional, but the discrete part is the significant zero-dimensional part. At the other extreme, the «-dimensional theory is just the Plancherel formula, which we write

Construction of units in integral group rings of finite nilpotent groups

Abstract: Let U be the group of units of the integral group ring of a finite group G. We give a set of generators of a subgroup B of U. This subgroup is of finite index in U if G is an odd nilpotent group. We also give an example of a 2-group such that B is of infinite index in U.