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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 Serre's modularity conjecture ： ウィキペディア英語版 Serre's modularity conjecture In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form, and a stronger version of his conjecture specifies the weight and level of the modular form. It was proved by Chandrashekhar Khare in the level 1 case,〔.〕 in 2005 and later in 2008 a proof of the full conjecture was worked out jointly by Khare and Jean-Pierre Wintenberger.〔 and .〕 ==Formulation== The conjecture concerns the absolute Galois group $G\_\backslash mathbb$ of the rational number field $\backslash mathbb$. Let $\backslash rho$ be an absolutely irreducible, continuous, two-dimensional representation of $G\_\backslash mathbb$ over a finite field $F\; =\; \backslash mathbb\_$. :$\backslash rho:\; G\_\backslash mathbb\; \backslash rightarrow\; \backslash mathrm\_2(F).\backslash$ Additionally, assume $\backslash rho$ is odd, meaning the image of complex conjugation has determinant -1. To any normalized modular eigenform :$f\; =\; q+a\_2q^2+a\_3q^3+\backslash cdots\backslash$ of level $N=N(\backslash rho)$, weight $k=k(\backslash rho)$, and some Nebentype character :$\backslash chi\; :\; \backslash mathbb/N\backslash mathbb\; \backslash rightarrow\; F^$ *\ , a theorem due to Shimura, Deligne, and Serre-Deligne attaches to $f$ a representation :$\backslash rho\_f:\; G\_\backslash mathbb\; \backslash rightarrow\; \backslash mathrm\_2(\backslash mathcal),\backslash$ where $\backslash mathcal$ is the ring of integers in a finite extension of $\backslash mathbb\_\backslash ell$. This representation is characterized by the condition that for all prime numbers $p$, coprime to $N\backslash ell$ we have :$\backslash operatorname(\backslash rho\_f(\backslash operatorname\_p))=a\_p\backslash$ and :$\backslash det(\backslash rho\_f(\backslash operatorname\_p))=p^\; \backslash chi(p).\backslash$ Reducing this representation modulo the maximal ideal of $\backslash mathcal$ gives a mod $\backslash ell$ representation $\backslash overline$ of $G\_\backslash mathbb$. Serre's conjecture asserts that for any $\backslash rho$ as above, there is a modular eigenform $f$ such that :$\backslash overline\; \backslash cong\; \backslash rho$. The level and weight of the conjectural form $f$ are explicitly calculated in Serre's article. In addition, he derives a number of results from this conjecture, among them Fermat's Last Theorem and the now-proven Taniyama–Weil (or Taniyama–Shimura) conjecture, now known as the modularity theorem (although this implies Fermat's Last Theorem, Serre proves it directly from his 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「Serre's modularity 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.