Distribution of almost division points
type: Item
instance of: scholarly article
title: Distribution of almost division points
zbMATH Open document ID: 0972.11053
DOI: 10.1215/S0012-7094-00-10313-4
author: Shou-Wu Zhang
published in: Duke Mathematical Journal
publication date: 2000-07-24T00:00:00Z
review text: In the paper under review, the author proves the following result, generalising earlier work [see \textit{S. Zhang}, Doc. Math., J. DMV, ICM Berlin 1998, Vol.~II, 217-225 (1998; Zbl 0912.14008); Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11034)]. Let \(A\) be an abelian variety over a number field \(K\), and let \((x_n)\) be a sequence of distinct points in \(A(\overline{K})\). Assume this is a sequence of almost division points, meaning that \[ \lim_{n \to \infty} \sup_{\sigma \in G} \|x_n^\sigma - x_n\|= 0, \] where \(G\) is the absolute Galois group of \(K\) and \(\|\cdot \|\) is the square root of some Néron-Tate height on \(A\). Then, replacing \((x_n)\) by a subsequence, there is an abelian subvariety \(C\) of \(A\), a point \(b \in A({\mathbb C})/C({\mathbb C})\) and a finite subset \(T\) of the torsion subgroup of \(A/C\) such that the measures \(\delta x_n^G\) (probability measure uniformly distributed over the points in the orbit \(x_n^G\)) on \(A({\mathbb C})\) converge to the measure \(d\mu = |T|^{-1} \sum_{t \in T} \delta_{\pi^{-1}(b+t)}\), where \(\pi : A \to A/C\) and \(\delta_{\pi^{-1}(b+t)}\) is the \(C({\mathbb C})\)-invariant probability measure supported on the fiber \(\pi^{-1}(b+t)\). As an application, the author proves that for a subvariety \(X\) of \(A\) that is not a translate of an abelian subvariety, the set of points in \(X(\overline{K})\) that are sufficiently near to \(A(K) \otimes {\mathbb R}\) (with respect to a Néron-Tate height pairing on \(A(\overline{K}) \otimes {\mathbb R}\)) is not Zariski-dense in \(X\).
reviewed by: Michael Stoll
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 1463078
zbMATH Keywords: abelian varieties over number fields
zbMATH Keywords: almost division points
zbMATH Keywords: equidistribution
MaRDI profile type: Publication
cites work: Limit distribution of small points on algebraic tori
cites work: Diophantine approximation on abelian varieties
cites work: Q4322758
cites work: Division points on semi-abelian varieties
cites work: Mordell-Lang plus Bogomolov
cites work: Q3704011
cites work: Equidistribution of small points
cites work: Positivity and discretion of algebraic points of curves
cites work: Equidistribution of small points on abelian varieties
cites work: Small points and Arakelov theory
full work available at URL: s0012-7094-00-10313-4
OpenAlex ID: W2149969967
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description: scientific article; zbMATH DE number 1463078