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Rational points of bounded height on compactifications of anisotropic tori. (English) Zbl 0890.14008

The main object of the paper under review is a smooth compactification \(X\) of an anisotropic torus \(T\) defined over a number field \(K\). The authors are interested in asymptotical distribution of rational points \(X(K)\) with respect to the anticanonical height. Their main result gives such an asymptotic expressed in geometric terms which is in full accordance with a conjecture by V. V. Batyrev and Yu. I. Manin [Math. Ann. 286, No. 1–3, 27–43 (1990; Zbl 0679.14008)] refined by E. Peyre [Duke Math. J. 79, No. 1, 101–218 (1995; Zbl 0901.14025)].
Note that recently the authors obtained the same asymptotics for a smooth compactification of an arbitrary torus \(T\) [J. Algebr. Geom. 7, 15–53 (1998; Zbl 0946.14009)].

MSC:

14G05 Rational points
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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