American Journal of Mathematics 123.1, February 2001
Contents
Walker, Mark E.

Weight one motivic cohomology and Ktheory
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Abstract:
We study a filtration of the algebraic Ktheory 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 K_{0}(X) is
studied, where
X is any quasiprojective variety.
(Pages 135.)
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 wavemaps equation
in n+1 dimensions, n=2,3, for initial data which is small in the
scaleinvariant homogeneous Besov space _{} X
_{} This result was proved in an earlier paper
by the author for n > 4.
(Pages 3777.)
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Dolgachev, Igor V.
Zhang, DeQi.

Coble rational surfaces
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Abstract:
A Coble surface is a smooth rational projective surface such that its
anticanonical linear system is empty while the antibicanonical 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 79114.)
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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 nonvanishing 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 115120.)
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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 lowerrank 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 Lfunctions.
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 Lfunctions on _{} X _{}.
(Pages 121161.)
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Dimca, Alexandru.
Saito, Morihiko.

Algebraic GaussManin systems and Brieskorn modules
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Abstract:
We study the algebraic GaussManin system and the algebraic
Brieskorn module associated to a polynomial mapping with isolated
singularities.
Since the algebraic GaussManin system does not contain any
information on the cohomology of singular fibers, we first construct a
nonquasicoherent sheaf which gives the cohomology of every fiber.
Then we study the algebraic Brieskorn module, and show that its
position in the algebraic GaussManin system is determined by a
natural map to quotients of local analytic GaussManin 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 twodimensional case we can describe the global structure of
the algebraic GaussManin system rather explicitly.
(Pages 163184.)
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