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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 Selmer group ： ウィキペディア英語版 Selmer group In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties. The Selmer group of an abelian variety ''A'' with respect to an isogeny ''f'' : ''A'' → ''B'' of abelian varieties can be defined in terms of Galois cohomology as :$\backslash mathrm^(A/K)=\backslash bigcap\_v\backslash mathrm(H^1(G\_K,\backslash mathrm(f))\backslash rightarrow\; H^1(G\_,A\_v())/\backslash mathrm(\backslash kappa\_v))$ where ''A''v() denotes the ''f''-torsion of ''A''v and $\backslash kappa\_v$ is the local Kummer map $B\_v(K\_v)/f(A\_v(K\_v))\backslash rightarrow\; H^1(G\_,A\_v())$. Note that $H^1(G\_,A\_v())/\backslash mathrm(\backslash kappa\_v)$ is isomorphic to $H^1(G\_,A\_v)()$. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have ''K''v-rational points for all places ''v'' of ''K''. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by ''f'' is finite due to the following exact sequence : 0 → ''B''(''K'')/''f''(''A''(''K'')) → Sel(f)(''A''/''K'') → Ш(''A''/''K'')() → 0. The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak Mordell–Weil theorem that its subgroup ''B''(''K'')/''f''(''A''(''K'')) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime ''p'' such that the ''p''-component of the Tate–Shafarevich group is finite. It is conjectured that the Tate–Shafarevich group is in fact finite, in which case any prime ''p'' would work. However, if (as seems unlikely) the Tate–Shafarevich group has an infinite ''p''-component for every prime ''p'', then the procedure may never terminate. Ralph Greenberg has generalized the notion of Selmer group to more general ''p''-adic Galois representations and to ''p''-adic variations of motives in the context of Iwasawa theory. == 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「Selmer group」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.