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

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.

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