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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 Néron–Tate height ： ウィキペディア英語版 Néron–Tate height In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell-Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate. ==Definition and properties== Néron defined the Néron–Tate height as a sum of local heights. Although the global Néron–Tate height is quadratic, the constituent local heights are not quite quadratic. Tate (unpublished) defined it globally by observing that the logarithmic height $h\_L$ associated to a symmetric invertible sheaf $L$ on an abelian variety $A$ is “almost quadratic,” and used this to show that the limit :$\backslash hat\; h\_L(P)\; =\; \backslash lim\_\backslash frac$ exists, defines a quadratic form on the Mordell-Weil group of rational points, and satisfies :$\backslash hat\; h\_L(P)\; =\; h\_L(P)\; +\; O(1),$ where the implied $O(1)$ constant is independent of $P$.〔Lang (1997) p.72〕 If $L$ is anti-symmetric, that is $()^$ *L=L, then the analogous limit :$\backslash hat\; h\_L(P)\; =\; \backslash lim\_\backslash frac$ converges and satisfies $\backslash hat\; h\_L(P)\; =\; h\_L(P)\; +\; O(1)$, but in this case $\backslash hat\; h\_L$ is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes $L^\; =\; (L\backslash otimes()^$ *L)\otimes(L\otimes()^ *L^) as a product of a symmetric sheaf and an anti-symmetric sheaf, and then :$\backslash hat\; h\_L(P)\; =\; \backslash frac12\; \backslash hat\; h\_(P)\; +\; \backslash frac12\; \backslash hat\; h\_\backslash quad\; \backslash hat\; h\_L(0)=0.$ The Néron–Tate height depends on the choice of an invertible sheaf on the abelian variety, although the associated bilinear form depends only on the image of $L$ in the Néron–Severi group of $A$. If the abelian variety $A$ is defined over a number field ''K'' and the invertible sheaf is symmetric and ample, then the Néron–Tate height is positive definite in the sense that it vanishes only on torsion elements of the Mordell-Weil group $A(K)$. More generally, $\backslash hat\; h\_L$ induces a positive definite quadratic form on the real vector space $A(K)\backslash otimes\backslash mathbb$. On an elliptic curve, the Néron-Severi group is of rank one and has a unique ample generator, so this generator is often used to define the Néron–Tate height, which is denoted $\backslash hat\; h$ without reference to a particular line bundle. (However, the height that naturally appears in the statement of the Birch–Swinnerton-Dyer conjecture is twice this height.) On abelian varieties of higher dimension, there need not be a particular choice of smallest ample line bundle to be used in defining the Néron–Tate height, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to the Poincaré line bundle on $A\backslash times\backslash hat\; A$, the product of $A$ with its dual. 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「Néron–Tate height」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.