Problems and results on the distribution of algebraic points on algebraic varieties
type: Item
instance of: scholarly article
title: Problems and results on the distribution of algebraic points on algebraic varieties
zbMATH Open document ID: 1194.11070
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://eudml.org/doc/10875
review text: This is a survey paper on the distribution of algebraic points on algebraic varieties. In the first section, the author considers varieties with many points. He quotes the main results on this topic and produces a collection of examples, especially of Fano varieties. He defines accumulators, he introduces conjectures of Batyrev and Manin, of Franke, Manin and Tschinkel. Next he pays some special attention to the case of the cubic threefold (also Theorem 11.10.11 of [\textit{E. Bombieri} and \textit{W. Gubler}, Heights in Diophantine geometry. New Mathematical Monographs~4. Cambridge: Cambridge University Press (2006; Zbl 1115.11034)]) which has important applications in additive number theory and in the theory of trigonometric sums. The next section deals with the Northcott property, and the last one with small points, including the theorems of \textit{F.~Amoroso} and \textit{S.~David} [J. Reine Angew. Math. 513, 145--179 (1999; Zbl 1011.11045)] and of \textit{Yu.~Bilu} [Duke Math. J. 89, No. 3, 465--476 (1997; Zbl 0918.11035)].
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 5620667
zbMATH Keywords: algebraic points
zbMATH Keywords: height
zbMATH Keywords: Weil absolute logarithmic height
zbMATH Keywords: Mahler's measure
zbMATH Keywords: circle method
zbMATH Keywords: accumulator
zbMATH Keywords: Fano varieties
zbMATH Keywords: K3 varieties
zbMATH Keywords: conjecture of Batyrev and Manin
zbMATH Keywords: Neron--Severi group
zbMATH Keywords: cubic threefold
zbMATH Keywords: Arakelov height
zbMATH Keywords: Schanuel's Theorem
zbMATH Keywords: conjecture of Franke, Manin and Tschinkel
zbMATH Keywords: Northcott property
zbMATH Keywords: small points
zbMATH Keywords: Lehmer conjecture
zbMATH Keywords: Bogomolov property
MaRDI profile type: Publication
OpenAlex ID: W2334269395
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DOI: 10.5802/JTNB.656
author: Enrico Bombieri
description: scientific article; zbMATH DE number 5620667