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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 Complex multiplication ： ウィキペディア英語版 Complex multiplication In mathematics, complex multiplication is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties ''A'' having ''enough'' endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of ''A'' is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. ==Example of the imaginary quadratic field extension== Consider an imaginary quadratic extension field $K=\backslash mathbb(\backslash sqrt)\backslash \; d>0$, which provides the typical example of complex multiplication. For the two periods $\backslash omega\_1,\backslash omega\_2$ of an elliptic function $f$ it is called to be of complex multiplication if there is an algebraic relation between $f(z)$ and $f(\backslash lambda\; z)$ for all $\backslash lambda$ in $K$ . Conversely, Kronecker conjectured that every abelian extension of $K$ would be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication, known as the ''Kronecker Jugendtraum'' and ''Hilbert's twelfth problem''. An example of an elliptic curve with complex multiplication is :$\backslash mathbb/\backslash mathbb()\backslash \; \backslash theta\backslash \; \backslash ;$ where Z() is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as :$Y^2\; =\; 4X^3\; -\; aX\backslash \; \backslash ,$ having an order 4 automorphism sending :$Y\; \backslash rightarrow-iY,\backslash \; \backslash \; \backslash \; \backslash \; X\; \backslash rightarrow-X$ in line with the action of ''i'' on the Weierstrass elliptic functions. More generally, consider the group of lattice L on the complex plane generated by $\backslash omega\_1,\backslash omega\_2$ . Then we define the Weierstrass function with a variable $z$ in $\backslash mathbb$ as follows: :$\backslash wp(z;L)=\backslash wp(z;\backslash omega\_1,\backslash omega\_2)=\backslash frac+\backslash sum\_\backslash left\backslash -\backslash frac\backslash right\backslash \}$ where :$g\_2\; =\; 60\backslash sum\_\; (m\backslash omega\_1+n\backslash omega\_2)^$ :$g\_3\; =140\backslash sum\_\; (m\backslash omega\_1+n\backslash omega\_2)^.$ Let $\backslash wp\text{'}$ be the derivative of $\backslash wp$. Then we obtain the isomorphism: :$z\backslash mapsto(1,\backslash wp(z),\backslash wp\text{'}(z))\; \backslash in\; \backslash mathbb^3$ which means the 1 to 1 correspondence between the complex torus group $\backslash mathbb/L$ and the elliptic curve :$E\; =\; \backslash left\backslash$ in the complex plane. This means that the ring of analytic automorphic group of $\backslash mathbb/L$ i.e., the ring of automorphisms of $E$, turn out to be isomorphic to the (subring of) integer rings of $K$ . In particular, assume $K\backslash subset\backslash mathbb$ and consider $L$ as an ideal of $K$ then $\backslash mathbb/L$ is agreed with the integer rings $\backslash mathfrak$ . Rewrite $\backslash tau=\backslash omega\_1/\backslash omega\_2\backslash \; \backslash text\backslash tau>0$ and $\backslash Delta(L)=g\_2(L)^3-27g\_3(L)^3$, then :$J(\backslash tau)=J(E)=J(L)=2^63^3g\_2(L)^3/\backslash Delta(L)\backslash \; .$ This means that the J-invariants of $E$ belong to the algebraic numbers of $K$ if $E$ has complex multiplication. 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「Complex multiplication」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.