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 [1] 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 which is 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. [2]
Manin's conjecture has been decided for special families of varieties, [3] but is still open in general.
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