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 be a Fano variety defined over a number field , let be a height function relative to the anticanonical divisor and assume that
is Zariski dense in . Then there exists a non-empty Zariski open subset
such that the counting function of -rational points of bounded height, defined by
for , satisfies
as Here
is the rank of the Picard group of and is a positive constant which later received a conjectural interpretation by Peyre.
Manin's conjecture has been proved for special families of varieties, but is still open in general.