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[1] in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.

Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface.

Conjecture

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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

 

as   Here   is the rank of the Picard group of   and   is a positive constant which later received a conjectural interpretation by Peyre.[2]

Manin's conjecture has been decided for special families of varieties,[3] but is still open in general.

References

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  1. ^ 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.
  2. ^ 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.
  3. ^ 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. Clay Mathematics Proceedings. Vol. 7. Providence, RI: American Mathematical Society. pp. 39–55. ISBN 978-0-8218-4307-9. MR 2362193. Zbl 1134.14017.