Bogomolov conjecture

In mathematics, the Bogomolov conjecture, named for Fedor Bogomolov, is the following 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. The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998.[1] 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, Zbl 0934.14013, doi:10.2307/120987.
  2. Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986

Further reading

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