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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bogomolov conjecture ： ウィキペディア英語版 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 $\backslash overline\; K$ denote the algebraic closure of ''K'', fix an embedding of ''C'' into its Jacobian variety ''J'', and let $\backslash hat\; h$ denote the Néron-Tate height on ''J'' associated to an ample symmetric divisor. Then there exists an $\backslash epsilon\; >\; 0$ such that the set :   is finite. Since $\backslash hat\; h(P)=0$ 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.〔.〕 Zhang proved the following generalization: Let ''A'' be an abelian variety defined over ''K'', and let $\backslash hat\; h$ be the Néron-Tate height on ''A'' associated to an ample symmetric divisor. A subvariety $X\backslash subset\; A$ 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 $\backslash epsilon\; >\; 0$ such that the set :   is not Zariski dense in ''A''. ==References== * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bogomolov conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.