## 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

### Citations:

Zbl 0861.14018; Zbl 0649.10014; Zbl 0991.11035; Zbl 0991.11030
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### References:

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