Root Numbers of Non-Abelian Twists of Elliptic Curves

We study the global root number of the complex L-function oftwists of elliptic curves over Q by real Artin representations.We obtain examples of elliptic curves over Q which, while nothaving any rational points of infinite order, conjecturallymust have points of infinite order over the fields for every cube-free m > 1. We describe analogousphenomena for elliptic curves over the fields , and in the towers and , where r 3 is prime.2000 Mathematics Subject Classification 11G40, 11G05.     

Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

Cohomology of Quantized Function Algebras at Roots of Unity

Let G be a simply-connected, semisimple algebraic group overk, an algebraically closed field of characteristic zero. Let O[G] be the quantized function algebra of G at a primitivelth root of unity , and let be the ‘restricted’ quantized function algebra at, a finite-dimensional k-algebra obtained from O[G] by factoringout a centrally generated ideal. It is known that is a Hopf algebra. We study the cohomology ring, a graded commutative algebra, and, for any finite-dimensional -module M, the -module . We prove that for sufficiently large l there isan isomorphism of graded algebras where each X_i is homogeneous of degree $2$, and $2N$ equalsthe number of roots associated to G. Moreover we show that inthis case is a finitely generated -module. We also show under much less restrictive conditions on l that continues to be a finitely generated graded commutativealgebra over which is a finitely generated module. 1991 Mathematics Subject Classification: 16W30,17B37, 17B56.     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Taut Foliations in Punctured Surface Bundles, II

Given a fibered 3-manifold M, we investigate exactly which boundaryslopes can be realized by perturbing fibrations along productdiscs. Since these perturbed fibrations cap off to give tautfoliations in the corresponding surgery manifolds, we obtainsurgery information. For example, recall that a knot k is saidto satisfy Property P if no finite surgery along k yields asimply-connected 3-manifold. We show that if a non-trivial fiberedknot k S³ fails to satisfy Property P, then necessarily k ishyperbolic with degeneracy slope . When k is hyperbolic and (respectively, ), we show that the only candidate for a counterexample to Property P is surgery coefficient (respectively, . 2000 Mathematical Subject Classification: primary57M25; secondary 57R30.     

On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

Quantum Deformations of Imaginary Verma Modules

We construct quantum deformations of imaginary Verma modulesover and show that, for generic q, imaginary Verma modules over can be deformed to those over the quantum group in such a way that the dimensions of the weightspaces are invariant under the deformation. We also prove thePBW theorem for with respect to the triangular decomposition induced from the root partitioncorresponding to the imaginary Verma modules. 1991 MathematicsSubject Classification: 17B67, 17B65, 17B10.     

Vinogradov's Integral and Bounds for the Riemann Zeta Function

The main result is an upper bound for the Riemann zeta functionin the critical strip: with A = 76.2 and B = 4.45, valid for 1 and |t| 3. The previousbest constant B was 18.5. Tools include a variant of the Korobov–Vinogradovmethod of bounding exponential sums, an explicit version ofT. D. Wooley's bounds for Vinogradov's integral, and explicitbounds for mean values of exponential sums over numbers withoutsmall prime factors, also using methods of Wooley. An auxiliaryresult is the exponential sum bound , where N is a positive integer, t is a real number, = log (t)/(logN) and 2000 Mathematical Subject Classification: primary 11M06, 11N05,11L15; secondary 11D72, 11M35.     

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G GL (n, ), we describe the coherent component Y of the moduli space of-stable McKay quiver representations. This is a not-necessarily-normaltoric variety that admits a projective birational morphism obtained by variation of GeometricInvariant Theory quotient. As a special case, this gives a newconstruction of Nakamura's G-Hilbert scheme Hilb^G that avoidsthe (typically highly singular) Hilbert scheme of |G|-pointsin . To conclude, we describe the toric fan of Y and hence calculate the quiver representationcorresponding to any point of Y.     

Asymptotic Singular Homology of a Complete Hyperbolic 3-Manifold of Finite Volume

Let V be a complete connected hyperbolic 3-manifold of finitevolume, with Liouville measure m, geodesic flow _t and Brownianmotion Z_t. Let be a smooth 1-form, closed in the cusps of V.Then the limit laws as t of under m and of are calculated, and seen to be Gaussian, and equal. The geodesic flow case isstudied via the Brownian case. 1991 Mathematics Subject Classification:60J65, 58F17, 51M10.     

The Second Bounded Cohomology of a Group Acting on a Gromov-Hyperbolic Space

Suppose a group G acts on a Gromov-hyperbolic space X properlydiscontinuously. If the limit set L(G) of the action has atleast three points, then the second bounded cohomology groupof is infinite dimensional. For example, if M is a complete, pinched negatively curved Riemannianmanifold with finite volume, then is infinite dimensional. As an application, we show that ifG is a knot group with GZ, then is infinite dimensional. 1991 Mathematics Subject Classification:primary 20F32; secondary 53C20, 57M25.     

On the Fredholm Alternative for the p-Laplacian in One Dimension

We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L¹(0,a) is a given function. The number _k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Let_{p p} x/a) denote the eigenfunction associated with ₁; then _p(k_p x/a) is the eigenfunction associated with _k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and f_t L¹(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t^* [0,1] such that (P) has a solutionfor f = f_t^*. Moreover, an appropriate choice of t_* yields asolution u with an arbitrarily large L¹(0,a)-norm which meansthat such f cannot be orthogonal to _pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto _p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t^* [0,1]such that (P) has a solution for f = f_t*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.     

The Distribution of {alpha}p Modulo One

In this paper the authors prove the following result. Let be an irrational number. Then for any > 0, there areinfinitely many prime numbers p such that | p | < p^–16/49+. The exponent improves on , which was obtained recently by thesecond author [Sci. China Ser. A 43 (2000) 703-721]. The resultis very close to the exponent , which can be obtained under the Generalized Riemann Hypothesis. Previous approaches to this problem have all used the same basicestimates for the trigonometric sums that arise. However, thepresent proof uses new bounds, which depend on the Kloostermansum and also on a counting problem in the geometry of numbers.In addition new techniques for the sieve method are applied.The most significant feature of the new approach is that, unlikeprevious methods, the exponential sum estimates remain non-trivialfor the exponent . This gives one hope for an unconditional result as good as that availableunder the Generalized Riemann Hypothesis. 2000 MathematicalSubject Classification: 11N36, 11J71.     

Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : W_r W, obtained as a composition of monoidal transformations,so that if X_r W_r denotes the strict transform of X W then: (1) the morphism : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.     

Weyl-Titchmarsh M-Function Asymptotics for Matrix-valued Schrodinger Operators

We explicitly determine the high-energy asymptotics for Weyl–Titchmarshmatrices corresponding to matrix-valued Schrödinger operatorsassociated with general self-adjoint m x m matrix potentials, where m N. More precisely,assume that for some N N and x₀R, for all c>x₀, and that x x₀ is a right Lebesgue point ofQ^(N–1). In addition, denote by I_m the mxm identity matrixand by C the open sector in thecomplex plane with vertex atzero, symmetry axis along the positive imaginary axis, and openingangle , with 0 < < . Then we prove the following asymptoticexpansion for any point M_+(z,x) of the unique limit point ora point of the limit disk associated with the differential expression in and a Dirichlet boundary condition at x=x₀: The expansion is uniform with respect to arg(z)for |z| in C and uniform in x as long as x varies in compactsubsets of R intersected with the right Lebesgue set of Q^(N–1).Moreover, the m x m expansion coefficients m_+,k(x) can be computedrecursively. Analogous results hold for matrix-valued Schrödinger operatorson the real line. 2000 Mathematics Subject Classification: 34E05,34B20, 34L40, 34A55.     

Multipliers on Banach spaces of functions on a locally compact Abelian group

Let E be a Banach space of functions on a locally compact Abeliangroup G satisfying certain conditions. It has been proved thatfor every bounded operator M on E commuting with translationsthere exists such that , where is a suitable subset of the group of the continuous morphismsfrom G into * and is a generalized Fourier transform of g defined on .     

Smooth Morphisms of Module Schemes

Let :A to B be a homomorphism of finite-dimensional algebrasover an algebraically closed field and the induced morphism of the associated module schemesfor any integer c 1. We prove that if the induced functor modB mod A is hom-controlled then the restriction of ^(c) to anyconnected component of is a composition of a smooth morphism followed by an immersion.Some new results on the types of singularities in the orbitclosures of module schemes are also proved. 2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.