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Rational Points of Bounded Height on Compactifications of Anisotropic Tori
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4 Rational Points of Bounded Height on
Compactifications of Anisotropic Tori
Victor V. Batyrev?
Universita?t-GHS-Essen, Fachbereich 6, Mathematik
Universita?tsstr. 3, 45141 Essen, FRG
e-mail: victor.batyrev@aixrs1.hrz.uni-essen.de
and
Yuri Tschinkel?
Harvard University, Department of Mathematics
1 Oxford Street, Cambridge, MA 02138, USA
e-mail: tschink@math.harvard.edu
Abstract
We investigate the analytic properties of the zeta-function associ-
ated with heights on equivariant compactifications of anisotropic tori
over number fields. This allows to verify conjectures about the distri-
bution of rational points of bounded height.
?Supported by DFG.
?Currently Junior Fellow of the Harvard Society of Fellows.
Contents
1 Toric varieties over arbitrary fields 5
1.1 Algebraic tori . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Compactifications of split tori . . . . . . . . . . . . . . . . . . 8
1.3 Compactifications of nonsplit tori . . . . . . . . . . . . . . . . 11
1.4 Algebraic tori over local and global fields . . . . . . . . . . . . 15
2 Heights and their Fourier transforms 20
2.1 Complexified local Weil functions and heights . . . . . . . . . 20
2.2 Fourier transforms of non-archimedian heights . . . . . . . . . 23
2.3 Fourier transforms of archimedian heights . . . . . . . . . . . 27
3 Characteristic functions of convex cones 29
4 Distribution of rational points 33
4.1 The method of Draxl . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 The meromorphic extension of ZΣ(?) . . . . . . . . . . . . . . 36
4.3 Rational points of bounded height . . . . . . . . . . . . . . . . 39
4.4 The residue at sj = 1 . . . . . . . . . . . . . . . . . . . . . . . 42
2
Introduction
In this paper we prove new results on the distribution ofK-rational points
of bounded height on algebraic varieties X defined over a number field K
[2, 9].
Let L = (L, ‖ · ‖v) be an ample metrized invertible sheaf on X wit
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