In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.
Their main conjecture is as follows. Let be a Fano variety defined over a number field , let be a height function which is relative to the anticanonical divisor and assume that is Zariski dense in . Then there exists a non-empty Zariski open subset such that the counting function of -rational points of bounded height, defined by
for , satisfies
Manin's conjecture has been decided for special families of varieties, but is still open in general.
- Franke, J.; Manin, Y. I.; Tschinkel, Y. (1989). "Rational points of bounded height on Fano varieties". Inventiones Mathematicae. 95 (2): 421–435. doi:10.1007/bf01393904. MR 0974910. Zbl 0674.14012.
- Peyre, E. (1995). "Hauteurs et mesures de Tamagawa sur les variétés de Fano". Duke Mathematical Journal. 79 (1): 101–218. doi:10.1215/S0012-7094-95-07904-6. MR 1340296. Zbl 0901.14025.
- Browning, T. D. (2007). "An overview of Manin's conjecture for del Pezzo surfaces". In Duke, William (ed.). Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany, June 20–24, 2005. Analytic number theory, Clay Math. Proc. Clay Mathematics Proceedings. 7. Providence, RI: American Mathematical Society. pp. 39–55. ISBN 978-0-8218-4307-9. MR 2362193. Zbl 1134.14017.