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Limit distribution of small points on algebraic tori. (English) Zbl 0918.11035
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 L. Szpiro, E. Ullmo, and S. Zhang [Invent. Math. 127, 337–347 (1997; Zbl 0991.11035)] and of 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 [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 S. Zhang [J. Am. Math. Soc. 8, 187–221 (1995; Zbl 0861.14018)] about small points on a closed subvariety of a torus follows easily from the main theorem of this paper.
Reviewer: P.Vojta (Berkeley)

MSC:
11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry
14G05 Rational points
11G25 Varieties over finite and local fields
11J68 Approximation to algebraic numbers
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