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.

Conjecture

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