Showing papers in "Crelle's Journal in 2001"

Existence of Primitive Divisors of Lucas and Lehmer Numbers

Abstract: We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

Wold-type decompositions and wandering subspaces for operators close to isometries

Abstract: For operators T satisfying certain inequalities we obtain a weak analog of the classical Wold-Kolmogorov decomposition theorem, representing T as a direct sum of a unitary operator and a shift operator acting in some Hilbert space of analytic functions The concept of a dual operator is introduced, which reflects relationships between shift operators acting in two Hilbert spaces of analytic functions dual to each other with respect to the Cauchy pairing Among different aspects of this duality we consider relationships between hereditary inequalities and properties of reproducing kernels

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Abstract: We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential “cup product” on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞-algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky

Abstract: We give a new proof of the theorem of Suslin-Voevodsky which shows that the Bloch-Kato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the Suslin-Voevodsky theorem to positive characteristic.

Extension of 2-forms and symplectic varieties

Abstract: This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included.

Identification of Berezin-Toeplitz Deformation Quantization

Abstract: We give a complete identification of the deformation quantization which was obtained from the Berezin- Toeplitz quantization on an arbitrary compact Kahler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose dassifying form is explicitly calculated. Its characteristic dass (which dassifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szego kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjostrand.

Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1

Abstract: Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional ``hypergeometric method'' for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with nZ 3, we show that the equation of the title possesses at most one solution in positive integers x; y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pade approximations to systems of binomial functions, together with new Chebyshev-like estimates for primes in arithmetic progressions and a variety of computational techniques.

Connected sums of constant mean curvature surfaces in Euclidean 3 space

Abstract: We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. Here nondegeneracy refers to the invertibility of the linearized mean curvature operator. This paper treats the simplest context for our result namely when the surfaces are compact with nonempty boundary, however the construction applies in the complete, noncompact setting as well. The surfaces we produce here are nondegenerate for generic choices of the free parameters in the construction.

On the number of solutions of the generalized Ramanujan-Nagell equation

Abstract: Let D1 and D2 be coprime positive integers and let k be an odd positive integer coprime with D1D2. We consider the Diophantine equation D1x2 + D2 = kn in the unknowns x≥1, n≥1. We give a necessary and sufficient condition on D1, D2 and k under which this equation has at most 2ω(k)−1 solutions where ω(k) denoted the number of distinct prime divisors of k. Thus, under a necessary and sufficient conditon, the equation has at most one solution whenever k is a prime. We also consider some related equations and we prove that the Diophantine equation x2+7=4yn has no solution in integers x≥1, y > 2 and n >1.

The Hall algebra of the category of coherent sheaves on the projective line

Abstract: To an abelian category A of homological dimension one satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth projective curve defined over a finite field, and observed analogies with quantum affine algebras. We recover here in an elementary way his results in the case when the curve is the projective line.

Generalized arithmetic intersection numbers

Abstract: We present an arithmetic intersection theory for hermitian line bundles on arithmetic surfaces, where the metrics are allowed to have logarithmic singularities at a finite set of points. Using this theory we show that the generalized arithmetic self-intersection number of the line bundle of modular forms equipped with its canonical metric equals ζ q (−1) + 2 · ζ ′ Q (−1) up to a trivial factor, where ζ Q ( s ) denotes the Riemann zeta function.

On the Chow motive of some moduli spaces

Abstract: We study the motive of moduli spaces of stable vector bundles over a smooth projective curve. We prove this motive lies in the category generated by the motive of the curve and we compute its class in the Grothendieck ring of the category of motives. As applications we compute the Poincare-Hodge polynomials and the number of points over a finite field and we study some conjectures on algebraic cycles on these moduli spaces.

On the Existence and Temperedness of Cusp Forms for SL(3,Z)

Abstract: We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$. As applications, we establish the Weyl asymptotic law for the discrete Laplace spectrum and prove that almost all of its cusp forms are tempered at infinity. The technique shows there are non-lifted cusp forms on $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$ as well as non-self-dual ones. A self-contained description of our proof for $SL_2({\Z})\backslash \U$ is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations.

Trace expansions and the noncommutative residue for manifolds with boundary

Abstract: For a pseudodifferential boundary operator A of order � 2 Z and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB s ), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Tr(AB s ) has a meromorphic extension to C with poles at the half-integers s = (n+� j)/2, j 2 N (possibly double for s < 0), and we prove that its residue at 0 equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Tr(A(B �) k ) in powersl/2 and log-powersl/2 log �, where the noncommutative residue equals the coefficient of the highest order log-power. There is a related expansion of Tr(Ae tB ). The paper will appear in Journal Reine Angew. Math. (Crelle's Journal). then is a holomorphic function of s for large Re s. We show that it extends to a mero- morphic function on the whole complex plane with at most double poles. Moreover, we prove that the noncommutative residue res(A) of A can be recovered as a residue in this

Nonvanishing of quadratic twists of modular l-functions and applications to elliptic curves

Abstract: If d is square-free or is a fundamental discriminant, then let χd = χD denote the Kronecker character for the quadratic field Q( √ d) whose fundamental discriminant is D. Throughout D shall denote a fundamental discriminant. The D-quadratic twist of F , denoted F ⊗ χD, is the newform corresponding to the twist of F by the character χD. In particular, if gcd(M,D) = 1, then (F ⊗ χD)(z) = ∑∞ n=1 χD(n)a(n)q n and

The classification of locally finite split simple Lie algebras

Abstract: One of the great early achievements in Lie theory is the classification of the finite dimensional simple complex Lie algebras by W. Killing and E. Cartan. More than 50 years later new aspects were added to this classification by the theory of Coxeter groups and the visualization of the classification in terms of Dynkin diagrams. Furthermore Serre’s description of the simple Lie algebras by generators and relations provided a direct way to construct the Lie algebras from the Cartan matrix corresponding to the choice of a root base, i.e., a system of simple roots.

On the automorphism groups of hyperbolic manifolds

Abstract: We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n e 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel space. These results complement earlier theorems of the authors on the possible dimensions of automorphism groups of domains in comlex space. The paper also contains a proof of our earlier result on characterizing $n$-dimensional hyperbolic complex manifolds with automorphism groups of dimension $\ge n^2+2$.

Pure Hodge structure on the $L_2$-cohomology of varieties with isolated singularities

Abstract: Cheeger, Goresky, and MacPherson conjectured in [CGM] an L2-de Rham theorem: that the intersection cohomology of a projective variety V is naturally isomorphic to the L2-cohomology of the incomplete manifold V −SingV , with metric induced by a projective embedding. The early interest in this conjecture was motivated in large part by the hope that one could then put a pure Hodge structure on the intersection cohomology of V and even extend the rest of the “Kahler package” ([CGM]) to this context. Saito ([S1,S2]) eventually established the Kahler package for intersection cohomology without recourse to L2-cohomology techniques. However, interest in L2-cohomology did not disappear with this result, since, among other things, L2-cohomology provides intrinsic geometric invariants of an arbitrary complex projective variety which are not apparent from the point of view of D-modules. For instance, L2− ∂-coholomology groups depend on boundary conditions ([PS]), which, as we show here, must be treated carefully in order to give the correct Hodge components for the L2-cohomology of a singular variety. A related fact is that for incomplete manifolds the pure Hodge structure and Lefschetz decompositions are not direct consequences of the Kahler condition as they are in the compact case. Indeed, the primary obstruction to obtaining a Hodge structure on the L2-cohomology is the following apparent technicality: on an incomplete Kahler manifold there are several potentially distinct definitions of a square integrable harmonic form. For example, a form h might be considered harmonic if dh = 0 = δh, or if ∂h = 0 = θh, or simply if ∆h = 0. Moreover there are further domain considerations: one imposes boundary conditions, which turn out to have no effect on cohomology in the case of d, but are crucial for ∂-cohomology. On a compact, or even complete manifold all these definitions of harmonics coincide, and one obtains the pure Hodge structure by decomposing harmonic forms into their (p, q) components. The (p, q) components are harmonic in the weakest sense they are in the kernel of ∆. The equality of the different notions of harmonic then allows one to realize these (p, q) components as spaces of both ∂ and d cohomology classes. The equivalence of the different definitions of harmonic is also required in order to obtain the Lefschetz decomposition. A local computation shows that interior product with the Kahler form preserves the kernel of ∆, but one requires the equivalence to see that this also induces an endomorphism on the L2-cohomology.