Limit distribution of small points on algebraic tori
type: Item
instance of: scholarly article
title: Limit distribution of small points on algebraic tori
zbMATH Open document ID: 0918.11035
DOI: 10.1215/S0012-7094-97-08921-3
published in: Duke Mathematical Journal
publication date: 1997-12-15T00:00:00Z
review text: Let \(A\) be an algebraic group over a number field \(k\). A sequence \(\{\alpha_i\}\) of points \(\alpha_i\in A(\overline k)\) is called strict if any proper algebraic subgroup of \(A\) contains \(\alpha_i\) for only finitely many values of \(i\). For a point \(\alpha\in A(\overline k)\), let \(\overline\delta_\alpha\) denote the measure \([k(\alpha):k]^{-1}\sum_{\sigma\:k(\alpha)\hookrightarrow \mathbb C} \delta_{\sigma(\alpha)}\) on \(A(\mathbb C)\), where \(\delta_x\) denotes the Dirac measure at \(x\in A(\mathbb C)\). We say that a sequence \(\{\mu_i\}\) of measures weakly converges to a measure \(\nu\) if \((f,\mu_i)\to(f,\nu)\) for all bounded continuous real-valued functions \(f\). This paper is inspired by the work of \textit{L. Szpiro, E. Ullmo}, and \textit{S. Zhang} [Invent. Math. 127, 337--347 (1997; Zbl 0991.11035)] and of \textit{S. Zhang} [Ann. Math. (2) 147, 159--165 (1998; Zbl 0991.11030)]. They showed that if \(A\) is an abelian variety, and if a strict sequence \(\{\alpha_i\}\) in \(A(\overline k)\) has heights \(h(\alpha_i)\to 0\), then the measures \(\overline\delta_{\alpha_i}\) weakly converge to the normalized Haar measure on \(A(\mathbb C)\). The present paper extends this result to tori: if \(A=\mathbb G_{\text{m}}^N\) and if \(k=\mathbb Q\), then for any strict sequence \(\{\alpha_i\}\) with heights \(h(\alpha_i)\to 0\), the measures \(\overline\delta_{\alpha_i}\) converge weakly to the normalized Haar measure on the closure (in the complex topology) of the torsion subgroup in \(A(\mathbb C)\). Note that the earlier result on abelian varieties remains true when stated in this form. The method of proof is to reduce to the case \(N=1\), where it can be shown by extensions of the author's earlier work [\textit{Yu. Belotserkovskii}, Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1988, No. 3, 111--115 (1988; Zbl 0649.10014)]. The final section of the paper shows that the theorem of \textit{S. Zhang} [J. Am. Math. Soc. 8, 187--221 (1995; Zbl 0861.140…
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 11J68
zbMATH DE Number: 1089061
zbMATH Keywords: equidistribution
zbMATH Keywords: small points
zbMATH Keywords: algebraic tori
zbMATH Keywords: strict sequence
zbMATH Keywords: weakly convergent measures
zbMATH Keywords: Haar measure
author: Yuri Bilu
reviewed by: Paul Vojta
MaRDI profile type: Publication
cites work: Q5560061
cites work: Q3789654
cites work: Q4853852
cites work: On a question of Lehmer and the number of irreducible factors of a polynomial
cites work: On the distribution of roots of polynomials
cites work: Q3309989
cites work: Exponential diophantine equations
cites work: Sur un théorème de M. Langevin
cites work: Q3998356
cites work: Solving algebraic equations in roots of unity.
cites work: Betti numbers of congruence groups. (Appendix: On representations of compact \(p\)-adic groups by Ze'ev Rudnick)
cites work: Equations in roots of unity
cites work: Heights of algebraic points lying on curves or hypersurfaces
cites work: Equidistribution of small points
cites work: Algebraic Numbers Close to Both 0 and 1
cites work: Positive Line Bundles on Arithmetic Varieties
cites work: Equidistribution of small points on abelian varieties
full work available at URL: s0012-7094-97-08921-3
OpenAlex ID: W2013963605
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description: scientific article; zbMATH DE number 1089061