In mathematics, the Bogomolov conjecture, named for Fedor Bogomolov, is the following statement:
Let C be an algebraic curve of genus g at least two defined over a number field K, let
K
¯
denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let
h
^
denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an
ϵ
>
0
such that the set
{
P
∈
C
(
K
¯
)
:
h
^
(
P
)
<
ϵ
}
is finite.
Since
h
^
(
P
)
=
0
if and only if P is a torsion point, the Bogomolov conjecture generalises the Manin-Mumford conjecture. The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. Zhang proved the following generalization:
Let A be an abelian variety defined over K, and let
h
^
be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety
X
⊂
A
is called a torsion subvariety if it is the translate of an abelian subvariety of A by a torsion point. If X is not a torsion subvariety, then there is an
ϵ
>
0
such that the set
{
P
∈
X
(
K
¯
)
:
h
^
(
P
)
<
ϵ
}
is not Zariski dense in
X.
