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 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 $\epsilon >0$ such that the set

\{P\in C({\overline {K}}):{\hat {h}}(P)<\epsilon \}

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 using Arakelov theory in 1998.^[1]^[2]

Generalization

In 1998, Zhang proved the following generalization:^[2]

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

\{P\in X({\overline {K}}):{\hat {h}}(P)<\epsilon \}

is not Zariski dense in X.

References

^ 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, Zbl 0934.14013.
^ ^a ^b Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986

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.