On the number of points of bounded height on arithmetic projective spaces
type: Item
instance of: scholarly article
title: On the number of points of bounded height on arithmetic projective spaces
zbMATH Open document ID: 0991.11036
author: Carlo Gasbarri
published in: Manuscripta Mathematica
publication date: 2000-01-31T00:00:00Z
review text: Let \(K\) be a number field and \(\mathcal{O}_K\) its ring of integers. Let \(\overline{E}\) be a Hermitian vector bundle over \(\text{Spec}(\mathcal{O}_K)\) of rank \(N\) and \(\mathbb{P}(\overline{E})\) its associated projective bundle. The first result of the paper uses Arakelov theory to prove an analogue of Schanuel's theorem on the number of points of bounded height on in a projective space over a number field [\textit{S. Schanuel}, Bull. Soc. Math. Fr. 107, 433-449 (1979; Zbl 0428.12009)]. Namely, an explicit upper bound depending on \(N\) and \(K\) for the number of points of \(\mathbb{P}(\overline{E})(K)\) whose normalized Arakelov height is at most \(T\) as \(T\to\infty\). Three applications of this result are given. In the first one, there is an estimate for the number of hyperplanes in \(\mathbb{P}(\overline{E})\) whose normalized Arakelov height is at most \(T\) containing a fixed linear subspace of \(\mathbb{P}(\overline{E})\) as \(T\to\infty\). The second one is an estimate for the number of hypersurfaces in \(\mathbb{P}(\overline{E})\) of degree at most \(T\) containing a fixed linear subspace as \(T\to\infty\). The third application is inspired by previous results [\textit{J. Franke, Y. Manin} and \textit{Y. Tschinkel}, Invent. Math. 95, 421-435 (1989; Zbl 0674.14012) and \textit{V. Batyrev} and \textit{Y. Manin}, Math. Ann. 286, 27-43 (1990; Zbl 0679.14008)] on the asymptotic estimate of points of bounded height on Fano varieties. Given the blow-up \(\widetilde{X}\) of \(\mathbb{P}^N_K\) at a linear subspace \(F\) of codimension two and \(\widetilde{T}\) the anticanonical line bundle of \(\widetilde{X}\), an estimate for the number of points in \(\widetilde{X}(K)\) of normalized Arakelov height with respect to \(\widetilde{T}\) at most \(T\) as \(T\to\infty\) is obtained.
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 1310838
zbMATH Keywords: Hermitian vector bundles
zbMATH Keywords: Arakelov heights
zbMATH Keywords: rational points
MaRDI profile type: Publication
full work available at URL: s002290050153
OpenAlex ID: W2007700374
DOI: 10.1007/S002290050153
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description: scientific article; zbMATH DE number 1310838