21,287 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
Rational points and non-anticanonical height functions
A conjecture of Batyrev and Manin predicts the asymptotic behaviour of
rational points of bounded height on smooth projective varieties over number
fields. We prove some new cases of this conjecture for conic bundle surfaces
equipped with some non-anticanonical height functions. As a special case, we
verify these conjectures for the first time for some smooth cubic surfaces for
height functions associated to certain ample line bundles.Comment: 16 pages; minor corrections; Proceedings of the American Mathematical
Society, 147 (2019), no. 8, 3209-322
Arithmetic progressions in binary quadratic forms and norm forms
We prove an upper bound for the length of an arithmetic progression
represented by an irreducible integral binary quadratic form or a norm form,
which depends only on the form and the progression's common difference. For
quadratic forms, this improves significantly upon an earlier result of Dey and
Thangadurai.Comment: 7 pages; minor revision; to appear in BLM
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