In mathematics, the Bogomolov conjecture, named for Fedor Bogomolov, is the following statement:
Let C be an algebraic curve of genus g at least two defined over a number field K, let K ¯ denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let 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 ∈ C ( K ¯ ) : h ^ ( P ) < ϵ } is finite.
Since 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. Zhang proved the following generalization:
Let A be an abelian variety defined over K, and let h ^ be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety X ⊂ 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 ∈ X ( K ¯ ) : h ^ ( P ) < ϵ } is not Zariski dense in
X.
