Abstract
We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batyrev, V. V. and Manin, Yu. I.: Sur le nombre de points rationnels de hauteur bornée des variétés algébriques, Math. Ann. 286 (1990), 27–43.
Batyrev, V. V. and Tschinkel, Yu.: Tamagawa numbers of polarized algebraic varieties, In: Nombre et répartition des points de hauteur bornée [14], pp. 299–340.
Birch, B. J.: Forms in many variables, Proc. London Math. Soc. 265A (1962), 245–263.
Chambert-Loir, A. and Tschinkel, Yu.: Torseurs arithmétiques et espaces fibrés, E-print, math.NT/9901006, 1999.
Davenport, H.: Cubic forms in sixteen variables, Proc. Roy. Soc. London Ser. A 272 (1963), 285–303.
Franke, J., Manin, Yu. I. and Tschinkel, Yu.: Rational points of bounded height on Fano varieties, Invent. Math. 95(2) (1989), 421–435.
Hartshorne, R.: AlgebraicGeometry, Grad. Texts in Math. 52, Springer, New York, 1977.
Hassett, B. and Tschinkel, Yu.: Geometry of equivariant compactifications of Gna, 1999.
Heath-Brown, R.: Cubic forms in 10 variables, Proc. London Math. Soc. 47(2) (1983), 225–257.
Hooley, C.: On nonary cubic forms, J. Reine Angew. Math. 386 (1988), 32–98.
Manin, Yu. I. and Panchishkin, A.: Number Theory I. Introduction to Number Theory, Springer, Berlin, 1995.
Mumford, D., Fogarty, J. and Kirwan F.: Geometric Invariant Theory, Ergeb. Math. Grenzgeb. (3) 34, Springer, Berlin, 1994.
Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), 101–218.
Peyre, E.: Nombre et répartition des points de hauteur bornée, Astérisque, No. 251, 1998.
Peyre, E.: Terme principal de la fonction zêta des hauteurs et torseurs universels, in Nombre et répartition des points de hauteur bornée [14], pp. 259–298.
16. Peyre, E.: Torseurs et méthode du cercle, Tech. report, Isaac Newton Institute for Mathematical Sciences, 1998.
Rademacher, H.: On the Phragmén-Lindelöf theorem and some applications, Math. Z. 72 (1959/60), 192–204.
Salberger, P.: Tamagawa measures on universal torsors and points of bounded height on Fano varieties, in Nombre et répartition des points de hauteur bornée [14], pp. 91–258.
Schanuel, S.: Heights in number fields, Bull. Soc. Math. France 107 (1979), 433–449.
Schmidt, W.: The density of integer points on homogeneous varieties, Acta Math. 154(3–4) (1985), 243-296.
Strauch, M. and Tschinkel, Yu.: Height zeta functions of toric bundles over flag varieties}, Selecta Math. 5(3) (1999), 325–396.
Tate, J. T.: Fourier analysis in number fields, and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chambert-Loir, A., Tschinkel, Y. Points of Bounded Height on Equivariant Compactifications of Vector Groups, I. Compositio Mathematica 124, 65–93 (2000). https://doi.org/10.1023/A:1002431622732
Issue Date:
DOI: https://doi.org/10.1023/A:1002431622732