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 formula_1 denote the algebraic closure of "K", fix an embedding of "C" into its Jacobian variety "J", and let formula_2 denote the Néron-Tate height on "J" associated to an ample symmetric divisor. Then there exists an formula_3 such that the set
Since formula_5 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 formula_2 be the Néron-Tate height on "A" associated to an ample symmetric divisor. A subvariety formula_7 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 formula_3 such that the set