21,748 research outputs found
On rings of integers generated by their units
We give an affirmative answer to the following question by Jarden and
Narkiewicz: Is it true that every number field has a finite extension L such
that the ring of integers of L is generated by its units (as a ring)? As a part
of the proof, we generalize a theorem by Hinz on power-free values of
polynomials in number fields.Comment: 15 page
Sums of units in function fields II - The extension problem
In 2007, Jarden and Narkiewicz raised the following question: Is it true that
each algebraic number field has a finite extension L such that the ring of
integers of L is generated by its units (as a ring)? In this article, we answer
the analogous question in the function field case.
More precisely, it is shown that for every finite non-empty set S of places
of an algebraic function field F | K over a perfect field K, there exists a
finite extension F' | F, such that the integral closure of the ring of
S-integers of F in F' is generated by its units (as a ring).Comment: 12 page
Moduli spaces of sheaves on K3 surfaces and Galois representations
We consider two K3 surfaces defined over an arbitrary field, together with a
smooth proper moduli space of stable sheaves on each. When the moduli spaces
have the same dimension, we prove that if the \'etale cohomology groups (with
Q_ell coefficients) of the two surfaces are isomorphic as Galois
representations, then the same is true of the two moduli spaces. In particular,
if the field of definition is finite and the K3 surfaces have equal zeta
functions, then so do the moduli spaces, even when the moduli spaces are not
birational.Comment: 16 pages. Improved proofs and exposition following referee's
suggestion
Preliminary galaxy extraction from DENIS images
The extragalactic applications of NIR surveys are summarized with a focus on
the ability to map the interstellar extinction of our Galaxy. Very preliminary
extraction of galaxies on a set of 180 consecutive images is presented, and the
results illustrate some of the pitfalls in attempting an homogeneous extraction
of galaxies from these wide-angle and shallow surveys.Comment: Invited talk at "The Impact of Large-Scale Near-IR Sky Surveys",
meeting held in Tenerife, Spain, April 1996. 10 pages LaTeX with style file
and 4 PS files include
Schanuel's theorem for heights defined via extension fields
Let be a number field, let be a nonzero algebraic number, and
let be the Weil height on the algebraic numbers. In response to a
question by T. Loher and D. W. Masser, we prove an asymptotic formula for the
number of with .
We also prove an asymptotic counting result for a new class of height
functions defined via extension fields of . This provides a conceptual
framework for Loher and Masser's problem and generalizations thereof.
Moreover, we analyze the leading constant in our asymptotic formula for Loher
and Masser's problem. In particular, we prove a sharp upper bound in terms of
the classical Schanuel constant.Comment: accepted for publication by Ann. Sc. Norm. Super. Pisa Cl. Sci., 201
Generalised divisor sums of binary forms over number fields
Estimating averages of Dirichlet convolutions , for some real
Dirichlet character of fixed modulus, over the sparse set of values of
binary forms defined over has been the focus of extensive
investigations in recent years, with spectacular applications to Manin's
conjecture for Ch\^atelet surfaces. We introduce a far-reaching generalization
of this problem, in particular replacing by Jacobi symbols with both
arguments having varying size, possibly tending to infinity. The main results
of this paper provide asymptotic estimates and lower bounds of the expected
order of magnitude for the corresponding averages. All of this is performed
over arbitrary number fields by adapting a technique of Daniel specific to
. This is the first time that divisor sums over values of binary forms
are asymptotically evaluated over any number field other than . Our
work is a key step in the proof, given in subsequent work, of the lower bound
predicted by Manin's conjecture for all del Pezzo surfaces over all number
fields, under mild assumptions on the Picard number
On Manin's conjecture for a certain singular cubic surface over imaginary quadratic fields
We prove Manin's conjecture over imaginary quadratic number fields for a
cubic surface with a singularity of type E_6.Comment: 16 pages. Both this article and arXiv:1304.3352 provide applications
of arXiv:1302.615
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