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.{{Citation |last=Ullmo |first=Emmanuel |title=Positivité et Discrétion des Points Algébriques des Courbes |journal=Annals of Mathematics |volume=147 |issue=1 |year=1998 |pages=167–179 |doi=10.2307/120987 |jstor=120987 | zbl=0934.14013 |arxiv=alg-geom/9606017 }}.{{Citation |last=Zhang|first=S.-W. |title=Equidistribution of small points on abelian varieties |journal=Annals of Mathematics |volume=147 |issue=1 |year=1998 |pages=159–165 |doi=10.2307/120986|jstor=120986 }}

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

: $\{ P \in X(\overline{K}) : \hat{h}(P) < \epsilon\}$ is not Zariski dense in X.

References

=Other sources=

{{cite book | author1-last=Chambert-Loir | author1-first=Antoine | chapter=Diophantine geometry and analytic spaces | pages=161–179 | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=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 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | location=Providence, RI | publisher=American Mathematical Society | isbn=978-1-4704-1021-6 | year=2013 }}