J-invariant

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 $SL(2, Z)$ 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 $j$ are modular, and in fact give all modular functions. Classically, the $j$ -invariant was studied as a parameterization of elliptic curves over $C$ , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).

While the $j$ -invariant can be defined purely in terms of certain infinite sums (see $g 2, g 3$ below), these can be motivated by considering isomorphism classes of elliptic curves. Every elliptic curve $E$ over $C$ is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of $C$ . 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 $1$ and some $τ$ in $H$ (where $H$ is the Upper half-plane). Conversely, if we define

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