1,076 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
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
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
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
Forms of differing degrees over number fields
Consider a system of polynomials in many variables over the ring of integers
of a number field . We prove an asymptotic formula for the number of
integral zeros of this system in homogeneously expanding boxes. As a
consequence, any smooth and geometrically integral variety satisfies the Hasse principle, weak approximation and the
Manin-Peyre conjecture, if only its dimension is large enough compared to its
degree.
This generalizes work of Skinner, who considered the case where all
polynomials have the same degree, and recent work of Browning and Heath-Brown,
who considered the case where . Our main tool is Skinner's number
field version of the Hardy-Littlewood circle method. As a by-product, we point
out and correct an error in Skinner's treatment of the singular integral.Comment: 23 pages; minor revision; to appear in Mathematik
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