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 $\overline K$ 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 $\epsilon > 0$ such that the set

: $\$   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 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 $\epsilon > 0$ such that the set

: $\$   is not Zariski dense in ''X''.

References

Other sources

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Further reading

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

Abelian varieties
Diophantine geometry
Conjectures that have been proved

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