翻訳と辞書
Words near each other
・ "O" Is for Outlaw
・ "O"-Jung.Ban.Hap.
・ "Ode-to-Napoleon" hexachord
・ "Oh Yeah!" Live
・ "Our Contemporary" regional art exhibition (Leningrad, 1975)
・ "P" Is for Peril
・ "Pimpernel" Smith
・ "Polish death camp" controversy
・ "Pro knigi" ("About books")
・ "Prosopa" Greek Television Awards
・ "Pussy Cats" Starring the Walkmen
・ "Q" Is for Quarry
・ "R" Is for Ricochet
・ "R" The King (2016 film)
・ "Rags" Ragland
・ ! (album)
・ ! (disambiguation)
・ !!
・ !!!
・ !!! (album)
・ !!Destroy-Oh-Boy!!
・ !Action Pact!
・ !Arriba! La Pachanga
・ !Hero
・ !Hero (album)
・ !Kung language
・ !Oka Tokat
・ !PAUS3
・ !T.O.O.H.!
・ !Women Art Revolution


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

Mordell-Weil theorem : ウィキペディア英語版
Mordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety ''A'' over a number field ''K'', the group ''A''(''K'') of ''K''-rational points of ''A'' is a finitely-generated abelian group, called the Mordell-Weil group. The case with ''A'' an elliptic curve ''E'' and ''K'' the rational number field Q is Mordell's theorem, answering a question apparently posed by Poincaré around 1908; it was proved by Louis Mordell in 1922.
==History==
The ''tangent-chord process'' (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group ''E''(Q)/2''E''(Q) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for ''E''(Q) to be finitely-generated; and it shows that the rank is finite. This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point on ''E''.
Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928. More abstract methods were required, to carry out a proof with the same basic structure. The second half of the proof needs some type of height function, in terms of which to bound the 'size' of points of ''A''(''K''). Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set of homogeneous coordinates. For an abelian variety, there is no ''a priori'' preferred representation, though, as a projective variety.
Both halves of the proof have been improved significantly, by subsequent technical advances: in Galois cohomology as applied to descent, and in the study of the best height functions (which are quadratic forms).

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「Mordell–Weil theorem」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.