Manin conjecture

Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface.(Picture kindly provided by U.Derenthal)

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.

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

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

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

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 974910. 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. 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.