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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 Weil pairing ： ウィキペディア英語版 Weil pairing In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing ''n'' of an elliptic curve ''E'', taking values in ''n''th roots of unity. More generally there is a similar Weil pairing between points of order ''n'' of an abelian variety and its dual. It was introduced by André Weil () for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function. ==Formulation== Choose an elliptic curve ''E'' defined over a field ''K'', and an integer ''n'' > 0 (we require ''n'' to be prime to char(''K'') if char(''K'') > 0) such that ''K'' contains a primitive nth root of unity. Then the ''n''-torsion on $E(\backslash overline)$ is known to be a Cartesian product of two cyclic groups of order ''n''. The Weil pairing produces an ''n''-th root of unity :$w(P,Q)\; \backslash in\; \backslash mu\_n$ by means of Kummer theory, for any two points $P,Q\; \backslash in\; E(K)()$, where $E(K)()=\backslash$ and $\backslash mu\_n\; =\; \backslash$. A down-to-earth construction of the Weil pairing is as follows. Choose a function ''F'' in the function field of ''E'' over the algebraic closure of ''K'' with divisor :$\backslash mathrm(F)=\; \backslash sum\_(P+k\backslash cdot\; Q)\; -\; \backslash sum\_\; (k\backslash cdot\; Q).$ So ''F'' has a simple zero at each point ''P'' + ''kQ'', and a simple pole at each point ''kQ'' if these points are all distinct. Then ''F'' is well-defined up to multiplication by a constant. If ''G'' is the translation of ''F'' by ''Q'', then by construction ''G'' has the same divisor, so the function ''G/F'' is constant. Therefore if we define :$w(P,Q):=\backslash frac$ we shall have an ''n''-th root of unity (as translating ''n'' times must give 1) other than 1. With this definition it can be shown that ''w'' is alternating and bilinear, giving rise to a non-degenerate pairing on the ''n''-torsion. The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of ''n''-torsion points) because the pairings for different ''n'' are not the same. However they do fit together to give a pairing ''T''ℓ(''E'') × ''T''ℓ(''E'') → ''T''ℓ(μ) on the Tate module ''T''ℓ(''E'') of the elliptic curve ''E'' (the inverse limit of the ℓ''n''-torsion points) to the Tate module ''T''ℓ(μ) of the multiplicative group (the inverse limit of ℓ''n'' roots of unity). 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「Weil pairing」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.