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 proved 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 denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an such that the set
Since 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.

Generalization

In 1998, Zhang proved the following generalization:
Let A be an abelian variety defined over K, and let be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety 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 such that the set