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 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 $$\overline K$$ denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let $$\hat h$$ denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an $$\epsilon > 0$$ such that the set


 * $$\{ P \in C(\overline{K}) : \hat{h}(P) < \epsilon\}$$   is finite.

Since $$\hat h(P)=0$$ 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.

Generalization
In 1998, Zhang proved the following generalization:

Let A be an abelian variety defined over K, and let $$\hat h$$ be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety $$X\subset A$$ 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 $$\epsilon > 0$$ such that the set


 * $$\{ P \in X(\overline{K}) : \hat{h}(P) < \epsilon\}$$   is not Zariski dense in X.