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In mathematics, Klein's ''j''-invariant or j function, regarded as a function of a complex variable ''τ'', is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that : Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). ==Definition== While the -invariant can be defined purely in terms of certain infinite sums (see below), these can be motivated by considering isomorphism classes of elliptic curves. Every elliptic curve over is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of . This is done by identifying opposite edges of each parallelogram in the lattice. It turns out that multiplying the lattice by complex numbers, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, and thus we can consider the lattice generated by and some in (where is the Upper half-plane). Conversely, if we define : then this lattice corresponds to the elliptic curve over defined by via the Weierstrass elliptic functions. Then the -invariant is defined as : where the ''modular discriminant'' is : It can be shown that is a modular form of weight twelve, and one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore , is a modular function of weight zero, in particular a meromorphic function invariant under the action of . As explained below, is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over and the complex numbers. 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「J-invariant」の詳細全文を読む スポンサード リンク
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