916 research outputs found

    On rings of integers generated by their units

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    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

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    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

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    Let kk be a number field, let θ\theta be a nonzero algebraic number, and let H()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 αk\alpha \in k with H(αθ)XH(\alpha \theta)\leq X. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of kk. 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

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    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

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    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

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    Consider a system of polynomials in many variables over the ring of integers of a number field KK. 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 XPKmX\subseteq \mathbb{P}_K^m 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 K=QK=\mathbb{Q}. 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

    Counting rational points on smooth cubic surfaces

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    We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.Comment: 11 pages, minor revisio

    Rational points of bounded height on general conic bundle surfaces

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    A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.Comment: 35 pages; final versio
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