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Manin conjecture : ウィキペディア英語版
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.
Their main conjecture is as follows.
Let V
be a Fano variety defined
over a number field K,
let H
be a height function which is relative to the anticanonical divisor
and assume that
V(K)
is Zariski dense in V.
Then there exists
a non-empty Zariski open subset
U \subset V
such that the counting function
of K-rational points of bounded height, defined by
:N_(B)=\#\
for B \geq 1,
satisfies
:N_(B) \sim c B (\log B)^,
as B \to \infty.
Here
\rho
is the rank of the Picard group of V
and 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 ==


抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Manin conjecture」の詳細全文を読む



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