• Source: 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



V


{\displaystyle V}


be a Fano variety defined
over a number field



K


{\displaystyle K}

,
let



H


{\displaystyle H}


be a height function which is 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 decided for special families of varieties, but is still open in general.




References

Kata Kunci Pencarian:

• Daftar masalah matematika yang belum terpecahkan
• Manin conjecture
• Yuri Manin
• Conjecture
• Arithmetic of abelian varieties
• List of conjectures
• André–Oort conjecture
• Faltings's theorem
• Fermat's Last Theorem
• Zilber–Pink conjecture
• List of unsolved problems in mathematics