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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 First Hurwitz triplet ： ウィキペディア英語版 First Hurwitz triplet In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, respectively the Klein quartic and the Macbeath surface). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces. ==Arithmetic construction== Let $K$ be the real subfield of $\backslash mathbb()$ where $\backslash rho$ is a 7th-primitive root of unity. The ring of integers of ''K'' is $\backslash mathbb()$, where $\backslash eta=2\backslash cos(\backslash tfrac)$. Let $D$ be the quaternion algebra, or symbol algebra $(\backslash eta,\backslash eta)\_$. Also Let $\backslash tau=1+\backslash eta+\backslash eta^2$ and $j\text{'}=\backslash tfrac(1+\backslash eta\; i\; +\; \backslash tau\; j)$. Let $\backslash mathcal\_\backslash mathrm=\backslash mathbb()()$. Then $\backslash mathcal\_\backslash mathrm$ is a maximal order of $D$ (see Hurwitz quaternion order), described explicitly by Noam Elkies (). In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in $\backslash mathbb()$, namely :$13=\backslash eta\; (\backslash eta\; +2)(2\backslash eta-1)(3-2\backslash eta)(\backslash eta+3),$ where $\backslash eta\; (\backslash eta+2)$ is invertible. Also consider the prime ideals generated by the non-invertible factors. The principal congruence subgroup defined by such a prime ideal ''I'' is by definition the group \mathrm :$\backslash mathcal^1\_\backslash mathrm(I)\; =\; \backslash ^1\; :\; x\; \backslash equiv\; 1\; \backslash pmod\}\backslash \},$ namely, the group of elements of reduced norm 1 in $\backslash mathcal\_\backslash mathrm$ equivalent to 1 modulo the ideal $I\backslash mathcal\_$. The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to PSL(2,R). Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a Fuchsian model, the quotient of the hyperbolic plane by one of these three Fuchsian groups. 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「First Hurwitz triplet」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.