American Journal of Mathematics

American Journal of Mathematics 123.1, February 2001

Contents

    Walker, Mark E.
  • Weight one motivic cohomology and K-theory
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    Abstract:
      We study a filtration of the algebraic K-theory spectrum of a smooth variety whose "layers" ought to give the motivic cohomology groups. The central result is a computation of the weight one layer of the filtration. The computations show that the homotopy groups of the weight one piece coincide with the weight one motivic cohomology groups, thus providing evidence that the filtration is correct. Additionally, a novel filtration of K0(X) is studied, where X is any quasi-projective variety. (Pages 1-35.) Abstract in TeX
    Tataru, Daniel.
  • On global existence and scattering for the wave maps equation
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    Abstract:
      We prove global existence and scattering for the wave-maps equation in n+1 dimensions, n=2,3, for initial data which is small in the scale-invariant homogeneous Besov space X This result was proved in an earlier paper by the author for n > 4. (Pages 37-77.) Abstract in TeX
    Dolgachev, Igor V.
    Zhang, De-Qi.
  • Coble rational surfaces
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    Abstract:
      A Coble surface is a smooth rational projective surface such that its anti-canonical linear system is empty while the anti-bicanonical linear system is nonempty. In this paper we shall classify Coble surfaces and consider the finiteness problem of the number of negative rational curves on it modulo automorphisms. (Pages 79-114.) Abstract in TeX
    Iosevich, Alex.
    Katz, Nets Hawk.
    Tao, Terry.
  • Convex bodies with a point of curvature do not have Fourier bases
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    Abstract:
      We prove that no smooth symmetric convex body W with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The nonsymmetric case was proven in a preprint by M. Kolountzakis.) This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if W can tile by translations. (Pages 115-120.) Abstract in TeX
    Stade, Eric.
  • Mellin transforms of Whittaker functions
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    Abstract:
      Using a known recursive formula for the class one principal series Whittaker function, we deduce a recursive formula for the multiple Mellin transform of this function. From the latter formula, we verify a conjecture of Goldfeld regarding the location of poles of our Mellin transform. We further express the residues at these poles in terms of Mellin transforms of lower-rank Whittaker functions. Our next result concerns the simplification of our Mellin transform under a certain restriction on the transform parameter. We show, by applying a change of variable to our above result on poles of the Mellin transform, that the transform reduces essentially to a transform under this restriction. We then demonstrate that, under further restriction of the Mellin transform parameter, this Mellin transform in fact reduces to a ratio of products of gamma functions. Our result proves a conjecture of Bump and Friedberg that is motivated by the theory of exterior square automorphic L-functions. Finally, we show that a certain Mellin transform of a product of two Whittaker functions (one on , and the other on reduces to a product of gamma functions. This last result verifies a conjecture of Bump regarding archimedean Euler factors of automorphic L-functions on X . (Pages 121-161.) Abstract in TeX
    Dimca, Alexandru.
    Saito, Morihiko.
  • Algebraic Gauss-Manin systems and Brieskorn modules
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    Abstract:
      We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic Gauss-Manin system does not contain any information on the cohomology of singular fibers, we first construct a nonquasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module, and show that its position in the algebraic Gauss-Manin system is determined by a natural map to quotients of local analytic Gauss-Manin systems, and its pole part by the vanishing cycles at infinity, comparing it with the Deligne extension. This implies for example a formula for the determinant of periods. In the two-dimensional case we can describe the global structure of the algebraic Gauss-Manin system rather explicitly. (Pages 163-184.) Abstract in TeX



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