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. 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 denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let denote the Néron-Tate height on J associated to an ample symmetric divisor. Then there exists an such that the set

  is finite.

Since 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.[1]

Generalization

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

Let A be an abelian variety defined over K, and let be the Néron-Tate height on A associated to an ample symmetric divisor. A subvariety 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 such that the set

  is not Zariski dense in X.

References

  1. Ullmo, E. (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.
  2. 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.

Further reading

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.