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 $\bar{K}$ 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 $ϵ>0$ such that the set

$\{P\in C(\bar{K}):\hat{h}(P)<ϵ\}$   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 $ϵ>0$ such that the set

$\{P\in X(\bar{K}):\hat{h}(P)<ϵ\}$   is not Zariski dense in X.

References

Other sources

• Template:Cite book

Further reading

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