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 \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. 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 \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 A.

References

  1. ↑ Ullmo, E. (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics 147 (1): 167–179, 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

Further reading

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