In mathematics, the j-invariant or j function is a modular function of weight zero for the special linear group SL ( 2 , Z ) {\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
j ( e 2 π i / 3 ) = 0 , j ( i ) = 1728 = 12 3 . {\displaystyle j{\big (}e^{2\pi i/3}{\big )}=0,\quad j(i)=1728=12^{3}.}
Rational functions of j {\displaystyle j} are modular, and in fact give all modular functions of weight 0. Classically, the j {\displaystyle j} -invariant was studied as a parameterization of elliptic curves over C {\displaystyle \mathbb {C} } , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).
Definition
The j-invariant can be defined as a function on the upper half-plane H = { τ ∈ C ∣ Im ( τ ) > 0 } {\displaystyle {\mathcal {H}}=\{\tau \in \mathbb {C} \mid \operatorname {Im} (\tau )>0\}} , by
j ( τ ) = 1728 g 2 ( τ ) 3 Δ ( τ ) = 1728 g 2 ( τ ) 3 g 2 ( τ ) 3 − 27 g 3 ( τ ) 2 = 1728 g 2 ( τ ) 3 ( 2 π ) 12 η ( τ ) 24 {\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 (\tau )^{24}}}}
with the third definition implying j ( τ ) {\displaystyle j(\tau )} can be expressed as a cube, also since 1728
= 12 3 {\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 Δ ( τ ) = g 2 ( τ ) 3 − 27 g 3 ( τ ) 2 = ( 2 π ) 12 η ( τ ) 24 {\displaystyle \Delta (\tau )=g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}=(2\pi )^{12}\,\eta (\tau )^{24}} , Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} , and modular invariants,
g 2 ( τ ) = 60 G 4 ( τ ) = 60 ∑ ( m , n ) ≠ ( 0 , 0 ) ( m + n τ ) − 4 {\displaystyle g_{2}(\tau )=60G_{4}(\tau )=60\sum _{(m,n)\neq (0,0)}\left(m+n\tau \right)^{-4}}
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