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 collaborators¹ 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 ${\displaystyle V}$ be a Fano variety defined over a number field ${\displaystyle K}$, let ${\displaystyle H}$ be a height function relative to the anticanonical divisor and assume that ${\displaystyle V(K)}$ is Zariski dense in ${\displaystyle V}$. Then there exists a non-empty Zariski open subset ${\displaystyle U\subset V}$ such that the counting function of ${\displaystyle K}$-rational points of bounded height, defined by

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

for ${\displaystyle B\geq 1}$, satisfies

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

as ${\displaystyle B\to \infty .}$ Here ${\displaystyle \rho }$ is the rank of the Picard group of ${\displaystyle V}$ and ${\displaystyle c}$ is a positive constant which later received a conjectural interpretation by Peyre.²

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