Showing papers in "Annals of Mathematics in 1995"

Modular elliptic curves and Fermat’s Last Theorem

Abstract: When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Ring-Theoretic Properties of Certain Hecke Algebras

Abstract: The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method going back to Mazur [M] allows one to show that these algebras are Gorenstein, but for the complete intersection property a new approach is required. The methods of this paper are related to those of Chapter 3 of [W2]. The methods of Section 3 of this paper are based on a previous approach of one of us (A.W.). We would like to thank Henri Darmon, Fred Diamond and Gerd Faltings for carefully reading the first version of this article. Gerd Faltings has also suggested a simplification of our argument as well as of the argument of Chapter 3 of [W2] and we would like to thank him for allowing us to reproduce these in the appendix to this paper. R. T. would like to thank A. W. for his invitation to collaborate and for sharing his many insights into the questions considered. R. T. would also like to thank Princeton University, Universite de Paris 7 and Harvard University for their hospitality during this collaboration. A. W. was supported by an NSF grant.

Decay of correlations

Abstract: *Dedicated to Micheline Ishay I would like to thank P. Boyland, L. Chierchia, V. Donnay, G. De Martino, C. Gole, J. L. Lebowitz, M. Lyubich, M. Rychlik, I. G. Schwarz, S. Vaienti and especially G. Gallavotti for helpful and enlightening discussions. Particularly warm thanks go to N. Chernov for carefully reading and finding a mistake in an early version; the present paper benefits from several improvements due to his sharp criticism. In addition, I am indebted to P. Collet and, most of all, M. Wojtkowski for introducing me to the magical world of cones. Finally, I thank J. Milnor, director of the Institute for Mathematical Sciences at Stony Brook University, where I was visiting during part of this work, and the Italian C.N.R.-GNFM for providing travel funds.

Double affine Hecke algebras and Macdonald's conjectures

Abstract: In this paper we prove the main results about the structure of double affine Hecke algebras announced in [C1], [C2]. The technique is based on the realization of these algebras in terms of Demazure-Lusztig operators [BGG], [D], [L2], [LS], [C3] and rather standard facts from the theory of affine Weyl groups. In particular, it completes the proof (partially published in [C2]) of the Macdonald scalar product conjecture (see [Ml], (12,6')), including the famous Macdonald constant-term conjecture (the q, t-case). We mainly follow the Opdam paper [0] where the Macdonald-Mehta conjectures in the degenerate (differential) case were deduced from certain properties of the Heckman-Opdam operators [HO] and the existence of the shift operators. Heckman's interpretation of these operators via the so-called Dunkl operators (see [He] and also [C5]) was important to our approach. We note that the HO operators are closely related to the so-called quantum many-body problem (Calogero, Sutherland, Moser, Olshanetsky, Perelomov), the conformal field theory (Knizhnik-Zamolodchikov equations), the harmonic analysis on symmetric spaces (Harish-Chandra, Helgason etc.), and (last but not the least) the classic theory of the hypergeometric functions. Establishing the connection between the difference counterparts of Heckman-Opdam operators introduced in [C4] and the Macdonald theory [Ml], [M2] including the construction of the difference shift operators is the main thrust of this paper. Once the connection is established it is not very difficult to calculate the scalar squares of the Macdonald polynomials and to prove the constant-term conjecture from his fundamental paper [M3]. To simplify the exposition, we discuss the reduced root systems only and impose the relation q = tk for k E Z+ (to avoid infinite products in the definition of Macdonald's pairing). The purpose of this work is to present a concrete application of the new technique. Arbitrary q, t can be handled in a

The geometry of symplectic energy

Abstract: "Non-Squeezing Theorem" which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form R2, x B2, where B2 is a 2-disc. This led to Hofer's discovery of symplectic capacities, which give a way of measuring the size of subsets in symplectic manifolds. Recently, Hofer found a way to measure the size (or energy) of symplectic diffeomorphisms by looking at the total variation of their generating Hamiltonians. This gives rise to a bi-invariant (pseudo-)norm on the group Ham(M) of compactly supported Hamiltonian symplectomorphisms of the manifold M. The deep fact is that this pseudo-norm is a norm; in other words, the only symplectomorphism on M with zero energy is the identity map. Up to now, this had been proved only for sufficiently nice symplectic manifolds, and by rather complicated analytic arguments. In this paper we consider a more geometric version of this energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of energy give rise to genuine norms on all symplectic manifolds. Roughly speaking, we show that if there were a symplectomorphism of M which had "too little" energy, one could embed a large ball into a thin cylinder M x

Extremal metrics of zeta function determinants on 4-manifolds

Abstract: In conformal geometry, the Sobolev inequality at a critical exponent has received much attention. In particular, the determination of the best constants has played a crucial role in the Yamabe problem. In dimension two the analogous problem deals with the Moser-Trudinger inequality: on a compact Riemann surface M2, there exists a constant c = c(M) so that f e47w2 < c(M) if f IVw12 < 1 and f w = 0. The connection of this inequality with geometry comes through the zeta functional determinant of the Laplacian as defined by Ray-Singer: for a Riemannian metric g, let 0 < A1 < A2 < ... be the spectrum

$L^\infty$ norms of eigenfunctions of arithmetic surfaces

Abstract: Here and elsewhere A ,l1o0 as A -xc is a basic problem of Quantum Chaos [S]. In any case almost nothing beyond (0.2) is known about 110,joc, when the curvature is negative (one can push the standard techniques and replace A1/4 by A1/4/ log A in this case). In this paper we use arithmetic techniques, in particular modular correspondences to obtain the first improvement in the exponent (0.2). We also obtain lower bounds on the L' norms in these arithmetic cases. In more detail, let A = (,b) be a quaternion division algebra over Q. A is linearly generated by 1, w, Q, wQ over Q and w2 = a, Q2 = b, wQ + Qw = 0. Here a, b E Z are square free and we will assume that a > 0. As usual the norm and trace are defined by N(a) = a-d, tr(a) = a + oY where if a = Xo + x1W + x2Q + X3WQ,

Holomorphic slices, symplectic reduction and multiplicities of representations

Abstract: I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

Mapping class groups are automatic

Abstract: Let S be a compact surface, possibly with the extra structure of an orientation or a finite set of distinguished points called punctures. The mapping class group of S is the group MCG(S) = Homeo(S)/ Homeo0(S), where Homeo(S) is the group of all homeomorphisms of S preserving the extra structure, and Homeo0(S) is the normal subgroup of all homeomorphisms isotopic to the identity through elements of Homeo(S). By convention each boundary component of S contains a puncture; in general, if a boundary component contains no puncture it may be collapsed to a puncture without changing the mapping class group. Given a group G, suppose that A = {g1, . . . , gk} is a finite set of generators and L is a set of words over the alphabet A, such that each element of G is represented by at least one word in L, and L is a regular language over A, i.e. one can check for membership in L with a finite automaton. The words in L representing a given group element can be thought of as normal forms for that element. Then L is an automatic structure for G if for any two words v, w ∈ L, one can check with a finite automaton whether the associated group elements v̄ and w̄ are equal, and whether they differ by a certain generator. A more geometric characterization of automaticity is given by the fellow traveller property , which says that there is a constant K such that for any v, w ∈ L, if d(v̄, w̄) ≤ 1, where d(v̄, w̄) is the word length of v̄−1w̄, then for any n ≥ 0, letting v(n) and w(n) be the prefixes of v̄ and w̄ of length n, then d(v̄(n), w̄(n)) ≤ K. A group G is automatic if it has an automatic structure. The theory of automatic groups is presented in [ECHLPT]. An automatic group has a quadratic isoperimetric inequality, and a quadratic time algorithm for the word problem, in addition to many other nice geometric and computational properties.

Finite energy solutions of the Yang-Mills equations in $\mathbb{R}^{3+1}$

Abstract: Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the size of the energy norm of the data. Since the energy norm is left invariant by the Yang-Mills flow the local solution is automatically extended to the entire space-time. Thus our results, which settle a problem stated in [Str], imply the well-known, fundamental, regularity result of Eardley and Moncrief [E-M]. That result, proved in the temporal gauge, requires a higher degree of smoothness for the data. Our main result, proved also in the temporal gauge, allows us to extend the concept of solutions to arbitrary finite energy initial data. The solutions are automatically unique in the class of solutions obtained by our procedure. Moreover, the global regularity proof given by Eardley and Moncrief depends in an essential way on the specific properties of the fundamental solution of the wave equation in the flat Minkowski space-time R3+1, namely the strong Huygens Principle. Indeed due mainly to this fact their proof does not seem to extend to general curved space-times. We have reasons to hope that the very different approach we take here will resolve this difficulty. The basic ingredients of our method are: 1. The introduction of appropriate local Coulomb gauges adapted to the causal structure of the equations. 2. An appropriate method of localizing the new space-time estimates for

Operators with Singular Continuous Spectrum: I. General Operators

Abstract: The Baire category theorem implies that the family, F, of dense sets G_ δ in fixed metric space, X, is a candidate for generic sets since it is closed under countable intersections; and if X is perfect (has no isolated point), then A ∈ F has uncountable intersections with any open ball in X.

All regulators of flat bundles are torsion

Abstract: I prove the Bloch conjecture:all secondary characteristic classes of flat bundles over complex projective varietes are torsion, except the first.

On topologically realizing modules over the Steenrod algebra

Abstract: The mod p cohomology of a space or spectrum X is a module over the mod p Steenrod algebra A. If X is a space, it has additional structure making it an unstable A-algebra, where we recall that an A-module M is unstable if, when p = 2, Sqi x = 0 for all i > IxI and x E M, and, when p > 2, 3ePix = 0 for all 2i + e > JxJ, e = 0 or 1, and x E M. Questions about realizing particular A-modules, algebras, and A-algebras as the cohomology of spaces or spectra are among the most fundamental in algebraic topology. The Hopf invariant one problem [A] and the problem of realizing polynomial algebras [DMW] are two famous examples of ongoing interest. If one regards the first of these as a problem about realizing finite A-modules, and the second of these as a problem about realizing rather large A-modules, it is the purpose of this paper to study the "middle ground". We begin with the following conjecture.

The diameter of the first nodal line of a convex domains

Abstract: The main goal of this paper is to prove that the first nodal line for the Dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain. More precisely, we locate the nodal line, to within a distance comparable to the inradius, near the zero of an ordinary differential equation, which is associated to the domain in a natural way. We also derive estimates for the first and second eigenvalues in terms of the corresponding eigenvalues of the ordinary differential equation and construct an approximate first eigenfunction. Two examples, a rectangle and a circular sector, illustrate the two extreme possibilities for the location of the nodal line. For the rectangle R = { (x, y) o 1, we have the second eigenfunction

Hard and soft packing radius theorems

Abstract: Singular spaces are playing an increasingly important role in Riemannian geometry. In particular this is true of the so-called Alexandrov spaces; i.e., finite-(Hausdorff) dimensional, complete, inner metric spaces with a lower curvature bound in the local triangle comparison sense. Interest in Alexandrov spaces is largely explained by the fact that the natural process of taking Gromov-Hausdorff limits is closed in Alexandrov geometry but not in Riemannian geometry. This simple observation has become a powerful tool in Riemannian geometry (see, for example, [FY], [GP2], or [GPW] and [Pe]). In this paper, we use a new means to obtain results in Riemannian geometry by studying Alexandrov spaces. The operation of forming spherical suspensions of positively curved spaces, an operation which is also closed in Alexandrov geometry but not in Riemannian geometry, will be applied to prove a differentiable sphere theorem which is unlike all previous solutions to the following basic problem.

Locally symmetric and rigid factors for complex manifolds via harmonic maps

Abstract: We obtain a complete description of the Lie algebra of complete holomorphic vector fields on the universal cover of a compact complex manifold with negative first Chern class. The main tools are an equivariance result we prove for harmonic maps and the rigidity theory for harmonic maps from Kahler manifolds to locally symmetric spaces.