Bogomolov conjecture
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. 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 [math]\displaystyle{ \overline K }[/math] denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let [math]\displaystyle{ \hat h }[/math] denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an [math]\displaystyle{ \epsilon \gt 0 }[/math] such that the set
- [math]\displaystyle{ \{ P \in C(\overline{K}) : \hat{h}(P) \lt \epsilon\} }[/math] is finite.
Since [math]\displaystyle{ \hat h(P)=0 }[/math] 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 in 1998.[1]
Generalization
In 1998, Zhang[2] proved the following generalization:
Let A be an abelian variety defined over K, and let [math]\displaystyle{ \hat h }[/math] be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety [math]\displaystyle{ X\subset A }[/math] 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 [math]\displaystyle{ \epsilon \gt 0 }[/math] such that the set
- [math]\displaystyle{ \{ P \in X(\overline{K}) : \hat{h}(P) \lt \epsilon\} }[/math] is not Zariski dense in X.
References
- ↑ Ullmo, E. (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics 147 (1): 167–179, doi:10.2307/120987.
- ↑ Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics 147 (1): 159–165, doi:10.2307/120986
Other sources
- Amini, Omid; Baker, Matthew; Faber, Xander, eds (2013). "Diophantine geometry and analytic spaces". 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. Contemporary Mathematics. 605. Providence, RI: American Mathematical Society. pp. 161–179. ISBN 978-1-4704-1021-6.
Further reading
 |  |
