Bogomolov conjecture

Let C be an algebraic curve of genus g at least two defined over a number field K, let ${\displaystyle {\overline {K}}}$ denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let ${\displaystyle {\hat {h}}}$ denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an ${\displaystyle \epsilon >0}$ such that the set

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

Since ${\displaystyle {\hat {h}}(P)=0}$ 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.[1]

In 1998, Zhang[2] proved the following generalization:

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

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

• The Manin-Mumford conjecture: a brief survey, by Pavlos Tzermias