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Statement

Let C be an algebraic curve of genus g at least two defined over a number field K, let K ¯ {\displaystyle {\overline {K}}} denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let h ^ {\displaystyle {\hat {h}}} denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an ϵ > 0 {\displaystyle \epsilon >0} such that the set

{ P ∈ C ( K ¯ ) : h ^ ( P ) < ϵ } {\displaystyle \{P\in C({\overline {K}}):{\hat {h}}(P)<\epsilon \}}   is finite.

Since h ^ ( P ) = 0 {\displaystyle {\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 1998.¹²

Generalization

In 1998, Zhang proved the following generalization:³

Let A be an abelian variety defined over K, and let h ^ {\displaystyle {\hat {h}}} be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety X ⊂ A {\displaystyle 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 ϵ > 0 {\displaystyle \epsilon >0} such that the set

{ P ∈ X ( K ¯ ) : h ^ ( P ) < ϵ } {\displaystyle \{P\in X({\overline {K}}):{\hat {h}}(P)<\epsilon \}}   is not Zariski dense in X.

Other sources

• Chambert-Loir, Antoine (2013). "Diophantine geometry and analytic spaces". In Amini, Omid; Baker, Matthew; Faber, Xander (eds.). Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Contemporary Mathematics. Vol. 605. Providence, RI: American Mathematical Society. pp. 161–179. ISBN 978-1-4704-1021-6. Zbl 1281.14002.

Further reading

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

References

1. Ullmo, Emmanuel (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics, 147 (1): 167–179, arXiv:alg-geom/9606017, doi:10.2307/120987, JSTOR 120987, Zbl 0934.14013. /wiki/Annals_of_Mathematics ↩

2. Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986 /wiki/Annals_of_Mathematics ↩

3. Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986 /wiki/Annals_of_Mathematics ↩