Kolmogorov's criterion: Difference between revisions
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In [[mathematics]], the '''Bogomolov conjecture''', named for [[Fedor Bogomolov]], is the following statement: | |||
Let ''C'' be an [[algebraic curve]] of [[genus (curve)|genus]] ''g'' at least two defined over a [[number field]] ''K'', let <math>\overline K</math> denote the [[algebraic closure]] of ''K'', fix an embedding of ''C'' into its [[Jacobian variety]] ''J'', and let <math>\hat h</math> denote the [[Néron-Tate height]] on ''J'' associated to an [[Ample divisor|ample symmetric divisor]]. Then there exists an <math>\epsilon > 0</math> such that the set | |||
: <math>\{ P \in C(\overline{K}) : \hat{h}(P) < \epsilon\}</math> is finite. | |||
Since <math>\hat h(P)=0</math> 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.<ref>{{Citation |last=Ullmo |first=E. |title={{lang|fr|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 | zbl=0934.14013 }}.</ref> | |||
Zhang<ref>{{Citation |last=Zhang|first=S.-W. |title={{lang|en|Equidistribution of small points on abelian varieties}} |journal=[[Annals of Mathematics]] |volume=147 |issue=1 |year=1998 |pages=159–165}}</ref> proved the following generalization: | |||
Let ''A'' be an [[abelian variety]] defined over ''K'', and let <math>\hat h</math> be the Néron-Tate height on ''A'' associated to an ample symmetric divisor. A [[subvariety]] <math>X\subset A</math> 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 <math>\epsilon > 0</math> such that the set | |||
: <math>\{ P \in X(\overline{K}) : \hat{h}(P) < \epsilon\}</math> is not [[Zariski dense]] in ''A''. | |||
==References== | |||
{{Reflist}} | |||
==Further reading== | |||
* [http://swc.math.arizona.edu/notes/files/99Tzermias.pdf The Manin-Mumford conjecture: a brief survey, by Pavlos Tzermias] | |||
[[Category:Abelian varieties]] | |||
[[Category:Diophantine geometry]] | |||
{{geometry-stub}} | |||
Revision as of 10:42, 21 January 2014
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
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 A.
References
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Further reading
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