Manin conjecture

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.

Conjecture

Their main conjecture is as follows. Let $V$ be a Fano variety defined over a number field $K$ , let $H$ be a height function which is relative to the anticanonical divisor and assume that $V(K)$ is Zariski dense in $V$ . Then there exists a non-empty Zariski open subset $U\subset V$ such that the counting function of $K$ -rational points of bounded height, defined by

N_{U,H}(B)=\#\{x\in U(K):H(x)\leq B\}

for $B\geq 1$ , satisfies

N_{U,H}(B)\sim cB(\log B)^{\rho -1},

as $B\to \infty .$ Here $\rho$ is the rank of the Picard group of $V$ and $c$ 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

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

[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.

[1]

[2]

[3]