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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 Vojta's conjecture ： ウィキペディア英語版 Vojta's conjecture In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in diophantine approximation theory, diophantine equations, arithmetic geometry, and logic. ==Statement of the Conjecture== Let $F$ be a number field, let $X/F$ be a non-singular algebraic variety, let $D$ be an effective divisor on $X$ with at worst normal crossings, let $H$ be an ample divisor on $X$, and let $K\_X$ be a canonical divisor on $X$. Choose Weil height functions $h\_H$ and $h\_$ and, for each absolute value $v$ on $F$, a local height function $\backslash lambda\_$. Fix a finite set of absolute values $S$ of $F$, and let $\backslash epsilon>0$. Then there is a constant $C$ and a non-empty Zariski open set $U\backslash subseteq\; X$, depending on all of the above choices, such that ::$\backslash sum\_\; \backslash lambda\_(P)\; +\; h\_(P)\; \backslash le\; \backslash epsilon\; h\_H(P)\; +\; C$ \quad\hbox P\in U(F). Examples: # Let $X=\backslash mathbb^N$. Then $K\_X\backslash sim\; -(N+1)H$, so Vojta's conjecture reads $\backslash sum\_\; \backslash lambda\_(P)\; \backslash le\; (N+1+\backslash epsilon)\; h\_H(P)\; +\; C$ for all $P\backslash in\; U(F)$. # Let $X$ be a variety with trivial canonical bundle, for example, an abelian variety, a K3 surface or a Calabi-Yau variety. Vojta's conjecture predicts that if $D$ is an effective ample normal crossings divisor, then the $S$-integral points on the affine variety $X\backslash setminus\; D$ are not Zariski dense. For abelian varieties, this was conjectured by Lang and proven by . # Let $X$ be a variety of general type, i.e., $K\_X$ is ample on some non-empty Zariski open subset of $X$. Then taking $S=\backslash emptyset$, Vojta's conjecture predicts that $X(F)$ is not Zariski dense in $X$. This last statement for varieties of general type is the Bombieri-Lang conjecture. 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「Vojta's 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.