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 $k$ be a number field, let $\theta$ be a nonzero algebraic number, and let $H(\cdot)$ 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 $\alpha \in k$ with $H(\alpha \theta)\leq X$. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. 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 $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ 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 $\chi$ 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 $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. 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