J-invariant

In mathematics, Felix Klein's j-invariant or j function is a modular function of weight zero for the special linear group ${\displaystyle \operatorname {SL} (2,\mathbb {Z} )}$ defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that

$${\displaystyle j{\big (}e^{2\pi i/3}{\big )}=0,\quad j(i)=1728=12^{3}.}$$

Rational functions of ${\displaystyle j}$ are modular, and in fact give all modular functions of weight 0. Classically, the ${\displaystyle j}$-invariant was studied as a parameterization of elliptic curves over ${\displaystyle \mathbb {C} }$, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).

The j-invariant can be defined as a function on the upper half-plane ${\displaystyle {\mathcal {H}}=\{\tau \in \mathbb {C} \mid \operatorname {Im} (\tau )>0\}}$, by

$${\displaystyle j(\tau )=1728{\frac {g_{2}(\tau )^{3}}{\Delta (\tau )}}=1728{\frac {g_{2}(\tau )^{3}}{g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}}}=1728{\frac {g_{2}(\tau )^{3}}{(2\pi )^{12}\,\eta ^{24}(\tau )}}}$$

with the third definition implying ${\displaystyle j(\tau )}$ can be expressed as a cube, also since 1728${\displaystyle {}=12^{3}}$. The function cannot be continued analytically beyond the upper half-plane due to the natural boundary at the real line.

The given functions are the modular discriminant ${\displaystyle \Delta (\tau )=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta ^{24}(\tau )}$, Dedekind eta function ${\displaystyle \eta (\tau )}$, and modular invariants,

$${\displaystyle g_{2}(\tau )=60G_{4}(\tau )=60\sum _{(m,n)\neq (0,0)}\left(m+n\tau \right)^{-4}}$$ $${\displaystyle g_{3}(\tau )=140G_{6}(\tau )=140\sum _{(m,n)\neq (0,0)}\left(m+n\tau \right)^{-6}}$$