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.

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 ⊂ V such that the counting function of K -rational points of bounded height, defined by

for B ≥ 1 , satisfies

as B → ∞ . Here ρ 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.