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 V {\displaystyle V} be a Fano variety defined over a number field K {\displaystyle K} , let H {\displaystyle H}
be a height function relative to the anticanonical divisor and assume that
V ( K ) {\displaystyle V(K)}
is Zariski dense in V {\displaystyle V} . Then there exists a non-empty Zariski open subset
U ⊂ V {\displaystyle U\subset V}
such that the counting function of K {\displaystyle K} -rational points of bounded height, defined by
N U , H ( B ) = # { x ∈ U ( K ) : H ( x ) ≤ B } {\displaystyle N_{U,H}(B)=\#\{x\in U(K):H(x)\leq B\}}
for B ≥ 1 {\displaystyle B\geq 1} , satisfies
N U , H ( B ) ∼ c B ( log B ) ρ − 1 , {\displaystyle N_{U,H}(B)\sim cB(\log B)^{\rho -1},}
as B → ∞ . {\displaystyle B\to \infty .}
Here
ρ {\displaystyle \rho }
is the rank of the Picard group of V {\displaystyle V}
and c {\displaystyle c}
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.
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