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 which is 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 decided for special families of varieties,^[3] but is still open in general.