Showing papers in "Duke Mathematical Journal in 2002"

Cuspidality of symmetric powers with applications

Abstract: The purpose of this paper is to prove that the symmetric fourth power of a cusp form on ${\rm GL}(2)$, whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the Ramanujan-Petersson and Sato-Tate conjectures. In particular, we establish the bound $q\sp {1/9}\sb v$ for unramified Hecke eigenvalues of cusp forms on ${\rm GL}(2)$. Over an arbitrary number field, this is the best bound available at present.

On level-zero representation of quantized affine algebras

Abstract: We study the properties of level-zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by T. Akasaka and the author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level-zero fundamental weight as an extremal weight is irreducible, and it is isomorphic to the affinization of an irreducible finite-dimensional module.

Rankin-Selberg $L$-functions in the level aspect

Abstract: Keywords: moments ; Rankin-Selberg convolution ; level aspect ; convexity-breaking Reference TAN-ARTICLE-2002-003doi:10.1215/S0012-7094-02-11416-1 Record created on 2008-11-14, modified on 2017-05-12

Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

Abstract: We establish borderline regularity for solutions of the Beltrami equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$ is a bounded measurable function, $\parallel\mu\parallel\sb \infty=k 0$. On the other hand, O. Lehto and T. Iwaniec showed that $q<1+k$ is not sufficient. In [2], the following question was asked: What happens for the borderline case $q=1+k$? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and two-weight estimates for the [26].

A polynomial bound in Freiman's theorem

Abstract: .Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s theorem.A fundamental result in the theory of set addition is Freiman’s theorem. Let A ⊂Z be a finite set of integers with small sumset; thus assume|A + A| <α|A|, (0.1)whereA + A = {x + y |x,y ∈ A} (0.2)and | · | denotes the cardinality. The factor αshould be thought of as a (possiblylarge) constant. Then Freiman’s theorem states that A is contained in a d-dimensionalprogression P, whered ≤ d(α) (0.3)and|P||A|≤ C(α). (0.4)(Precise definitions are given in Section 1.) Although this statement is very intuitive,there is no simple proof so far, and it is one of the deep results in additive numbertheory.G. Freiman’s book [Fr] on the subject is not easy to read, which perhaps explainswhy in earlier years the result did not get its deserved publicity. More recently, two

Riemannian manifolds with maximal eigenfunction growth

Abstract: On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L\sp 2$-normalized eigenfunctions $\{\phi\sb \lambda\}$ satisfy $\lvert \rvert\phi\sb \lambda\lvert \rvert\sb\infty\leq C\lambda\sp {(n-1)/2}$, where $-\Delta\phi\sb \lambda=\lambda\sp 2\phi\sb \lambda$. The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S\sp n$. But, of course, it is not sharp for many Riemannian manifolds, for example, for flat tori $\mathbb {R}\sp n/\Gamma$. We say that $S\sp n$, but not $\mathbb {R}\sp n/\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem that motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. Our main result is that such an $(M, g)$ must have a point $x$ where the set $\mathscr {L}\sb x$ of geodesic loops at $x$ has positive measure in $S\sp \ast\sb xM$. We show that if $(M, g)$ is real analytic, this puts topological restrictions on $M$; for example, only $M=S\sp 2$ or $M=\mathbb {R}P\sp 2$ (topologically) in dimension $2$ can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any $M$ fail to have maximal eigenfunction growth. In addition, we construct an example of $(M, g)$ for which $\mathscr {L}\sb x$ has positive measure for an open set of $x$ but which does not have maximal eigenfunction growth; thus, it disproves a naive converse to the main result.

From local to global deformation quantization of Poisson manifolds

Abstract: We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

Uniform semiclassical estimates for the propagation of quantum observables

Abstract: We prove here that the semiclassical asymptotic expansion for the propagation of quantum observables, for C\sp ∞-Hamiltonians growing at most quadratically at infinity, is uniformly dominated at any order by an exponential term whose argument is linear in time. In particular, we recover the Ehrenfest time for the validity of the semiclassical approximation. This extends the result proved in [BGP]. Furthermore, if the Hamiltonian and the initial observables are holomorphic in a complex neighborhood of the phase space, we prove that the quantum observable is an analytic semiclassical observable. Other results about the large time behavior of observables with emphasis on the classical dynamic are also given. In particular, precise Gevrey estimates are established for classically integrable systems.

A Darboux theorem for Hamiltonian operators in the formal calculus of variations

Abstract: We prove a formal Darboux-type theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras $\mathfrak {g}$ concentrated in degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are parametrized by a 2-groupoid that we call the Deligne 2-groupoid of $\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras have equivalent Deligne 2-groupoids.

Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants

Abstract: Let $k$ be a local field of characteristic not $2$, and let $G$ be the group of $k$-rational points of a connected reductive linear algebraic group defined over $k$ with a simple derived group of $k$-rank at least $2$. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of $G$. These bounds turn out to be optimal for ${\rm SL}\sb n(k), n\geq 3$, and ${\rm Sp}\sb {2n}(k),n\geq 2$. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple $G$.

Potential estimates for a class of fully nonlinear elliptic equations

Abstract: We study the pointwise properties of $k$-subharmonic functions, that is, the viscosity subsolutions to the fully nonlinear elliptic equations $F_k[u]=0$, where $F_k[u]$ is the elementary symmetric function of order $k,1\leq k\leq n$, of the eigenvalues of $[D\sp 2u]$, $F_1[u]=\Delta u,F_n[u]=\det D^2u$. Thus $1$-subharmonic functions are subharmonic in the classical sense; $n$-subharmonic functions are convex. We use a special capacity to investigate the typical questions of potential theory: local behaviour, removability of singularities, and polar, negligible, and thin sets, and we obtain estimates for the capacity in terms of the Hausdorff measure. We also prove the Wiener test for the regularity of a boundary point for the Dirichlet problem for the fully nonlinear equation $F_k[u]=0$. The crucial tool in the proofs of these results is the Radon measure $F_k[u]$ introduced recently by N. Trudinger and X.-J. Wang for any $k$-subharmonic $u$. We use ideas from the potential theories both for the complex Monge-Ampere and for the $p$-Laplace equations.

The Cramer-Rao inequality for star bodies

Abstract: Associated with each body $K$ in Euclidean $n$-space $\mathbb {R}\sp n$ is an ellipsoid $\Gamma\sb 2K$ called the Legendre ellipsoid of $K$. It can be defined as the unique ellipsoid centered at the body's center of mass such that the ellipsoid's moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body $K\subset\mathbb {R}\sp n$ is a new ellipsoid $\Gamma\sb {-2}K$ that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn-Minkowski theory, while the new ellipsoid $\Gamma\sb {-2}K$ is the corresponding object of the Brunn-Minkowski theory. The present paper has two aims. The first is to show that the domain of $\Gamma\sb {-2}$ can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If $K$ is a star-shaped set, then $\Gamma\sb {-2}K\subset\Gamma\sb 2K$ with equality if and only if $K$ is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory–the Cramer-Rao inequality.

Multiplicités modulaires et représentations de ${\rm GL}\sb 2(\mathbf {Z}\sb p)$ et de ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ en $\ell=p$. Appendice par Guy Henniart. Sur l'unicité des types pour ${\rm GL}\sb 2$

Abstract: We formulate a conjecture giving a link between the various rings parametrizing the $2$-dimensional potentially semistable $p$-adic representations of ${\rm Gal}(\overline {\mathbf {Q}}\sb p/\mathbf {Q}\sb p)$ with Hodge-Tate weights $(0,k-1)(k\in \mathbf {Z},1

Accretive system Tb-theorems on nonhomogeneous spaces

Abstract: We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderon-Zygmund operator on L2(μ)$. We do not assume any kind of doubling condition on the measure $\mu$, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L\sp ∞. Thus we answer positively a question of Christ as to whether the L\sp ∞-assumption can be replaced by a BMO assumption. We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calderon-Zygmund operators with respect to very bad measures.

Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation

Abstract: The main results of this paper are concerned with the "bad" behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class $C\sp 2$ in any Sobolev space.

Fourier coefficients of modular forms on G2

Abstract: We develop a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ.

Gluing tight contact structures

Abstract: We prove gluing theorems for tight contact structures. As special cases, we rederive gluing theorems due to V. Colin and S. Makar-Limanov and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus $4$ handlebody which becomes overtwisted after Legendrian $-1$ surgery and study certain Legendrian surgeries on $T\sp 3$.

A Kohno-Drinfeld theorem for quantum Weyl groups

Abstract: Let $\mathfrak {g}$ be a complex, simple Lie algebra with Cartan subalgebra $\mathfrak {h}$ and Weyl group $W$. In [MTL], we introduced a new, $W$-equivariant flat connection on $\mathfrak {h}$ with simple poles along the root hyperplanes and values in any finite-dimensional $\mathfrak {g}$-module $V$. It was conjectured in [TL] that its monodromy is equivalent to the quantum Weyl group action of the generalised braid group of type $\mathfrak {g}$ on $V$ obtained by regarding the latter as a module over the quantum group $U\sb \hbar\mathfrak {g}$. In this paper, we prove this conjecture for $\mathfrak {g}=\mathfrak {sl}\sb n$.

Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

Abstract: For any smooth complex projective variety $X$ and any smooth very ample hypersurface $Y\subset X$, we develop the technique of genus zero relative Gromov-Witten invariants of $Y$ in $X$ in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps which relates these relative invariants to the Gromov-Witten invariants of $X$ and $Y$. Given the Gromov-Witten invariants of $X$, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of $Y$ whose homology and cohomology classes are induced by $X$.

Hua-type integrals over unitary groups and over projective limits of unitary groups

Abstract: We discuss some natural maps from a unitary group ${\rm U}(n)$ to a smaller group ${\rm U}(n-m)$. (These maps are versions of the Livsic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces. (Values of the integrals are products of $\Gamma$-functions.) These integrals generalize Hua integrals. We construct inverse limits of unitary groups equipped with analogues of Haar measure and evaluate some integrals over these inverse limits.

Standard modules of quantum affine algebras

Abstract: We give a proof of the cyclicity conjecture of Akasaka and Kashiwara for simply laced types, via quiver varieties. We also get an algebraic characterization of the standard modules.

Localization for one dimensional, continuum, bernoulli-anderson models

Abstract: We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, continuum, Anderson-type models with singular distributions; in particular the case of a Bernoulli dis- tribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that non-reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs.

Sharp inequalities for functional integrals and traces of conformally invariant operators

Abstract: The intertwining operators $A\sb d=A\sb d(g)$ on the round sphere $(S\sp n,g)$ are the conformal analogues of the power Laplacians $\Delta\sp {d/2}$ on the flat $\mathbf {R}\sp n$. To each metric $\rho g$, conformally equivalent to $g$, we can naturally associate an operator $A\sb d(\rho g)$, which is compact, elliptic, pseudodifferential of order $d$, and which has eigenvalues $\lambda\sb j(\rho)$; the special case $d=2$ gives precisely the conformal Laplacian in the metric $\rho g$. In this paper we derive sharp inequalities for a class of trace functionals associated to such operators, including their zeta function $\sum\sp j\lambda\sp j(\rho)\sp {-s}$, and its regularization between the first two poles. These inequalities are expressed analytically as sharp, conformally invariant Sobolev-type (or log Sobolev type) inequalities that involve either multilinear integrals or functional integrals with respect to $d$-symmetric stable processes. New strict rearrangement inequalities are derived for a general class of path integrals.

Galois representations with conjectural connections to arithmetic cohomology

Abstract: In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the $\mod p$ cohomology of congruence subgroups of ${\rm SL}\sb n(\mathbb {Z})$ to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case $n=3$ in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.

Fundamental solutions for the tricomi operator

Abstract: In this paper we explicitly calculate fundamental solutions for the Tricomi operator, relative to an arbitrary point in the plane, and show that all such fundamental solutions originate from the hypergeometric function F(1/6,1/6;1;ζ)$ that is obtained when we look for homogeneous solutions to the reduced hyperbolic Tricomi equation.

Upper and lower bounds at s=1 for certain Dirichlet series with Euler product

Abstract: Estimates of the form $L^{(j)}(s,\mathscr{A})\ll_{\epsilon,j,\mathscr {D_A}}\mathscr {R}^\epsilon_{\mathscr {A}}$ in the range $|s-1|\ll 1/\log \mathscr {R_A}$ for general $L$-functions, where $\mathscr {R_A}$ is a parameter related to the functional equation of $L(s,\mathscr {A})$, can be quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates when the $L$-functions have Euler product of polynomial type and the Ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions. As a consequence, we obtain an upper bound of this type for every $L(s, \pi)$, where $\pi$ is an automorphic cusp form on ${\rm GL}(\mathbf {d},\mathbb {A}_K)$. We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form.

Special Lagrangian m-folds in ℂm with symmetries

Abstract: This the first in a series of papers on special Lagrangian submanifolds in ℂm. We study special Lagrangian submanifolds in ℂm with large symmetry groups, and we give a number of explicit constructions. Our main results concern special Lagrangian cones in ℂm invariant under a subgroup G in SU(m) isomorphic to U(1)m−2. By writing the special Lagrangian equation as an ordinary differential equation (ODE) in G-orbits and solving the ODE, we find a large family of distinct, G-invariant special Lagrangian cones on Tm−2 in ℂm. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models are needed to understand mirror symmetry and the Strominger-Yau-Zaslow (SYZ) conjecture.

Riemannian manifolds with uniformly bounded eigenfunctions

Abstract: The standard eigenfunctions $\phi_\lambda=e^{i\langle\lambda,x\rangle}$ on flat tori $\mathbb {R}^n/L$ have $L^\infty$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $^\infty$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians.

Geometric branched covers between generalized manifolds

Abstract: We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension.

Global asymptotics for multiple integrals with boundaries

Abstract: Under convenient geometric assumptions, the saddle-point method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points. The incidence of the Stokes phenomenon is related to the monodromy of the homology via generalized Picard-Lefschetz formulae and is quantified in terms of geometric indices of intersection. Exact remainder terms and the hyperasymptotics are then derived. A direct consequence is a numerical algorithm to determine the Stokes constants and indices of intersections. Examples are provided.