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 \overline 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 such that the set 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 such that the set

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