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 ${\displaystyle {\overline {K}}}$ 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.

Proof

The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang using Arakelov theory in 1998.^[1][2]

Generalization

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

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.