Bogomolov conjecture explained
In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proven by Emmanuel Ullmo and Shou-Wu Zhang in 1998 using Arakelov theory. A further generalization to general abelian varieties was also proved by Zhang in 1998.
Statement
Let C be an algebraic curve of genus g at least two defined over a number field K, let
denote the
algebraic closure of
K, fix an embedding of
C into its
Jacobian variety J, and let
denote the
Néron-Tate height on
J associated to an
ample symmetric divisor. Then there exists an
such that the set
\{P\inC(\overline{K}):\hat{h}(P)<\epsilon\}
is finite.
Since
if and only if
P is a
torsion point, the Bogomolov conjecture generalises the Manin-Mumford conjecture.
Proof
The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang using Arakelov theory in 1998.[1]
Generalization
In 1998, Zhang proved the following generalization:
Let A be an abelian variety defined over K, and let
be the Néron-Tate height on
A associated to an ample symmetric divisor. A
subvariety
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
such that the set
\{P\inX(\overline{K}):\hat{h}(P)<\epsilon\}
is not
Zariski dense in
X.
References
Other sources
- Book: Chambert-Loir . Antoine . Diophantine geometry and analytic spaces . 161–179 . Amini . Omid . Baker . Matthew . Faber . Xander . Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 . 1281.14002 . Contemporary Mathematics . 605 . Centre de Recherches Mathématiques Proceedings--> . Providence, RI . . 978-1-4704-1021-6 . 2013 .
Further reading
Notes and References
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