J-invariant explained

In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for special linear group 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\left(e ^{2\pi} \right) = 0 , j(i) = 1728 = 12 ^{3.}

Rational functions of are modular, and in fact give all modular functions of weight 0. Classically, the -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).

Definition

j(\tau) = 1728 (g ^{3} _{2 (\tau)})/(\Delta(\tau)) = 1728 (g ^{3} _{2 (\tau)})/(g ^{3} _{2 (\tau)} - 27 g ^{2} _{3 (\tau)}) = 1728 (g ^{3} _{2 (\tau)})/(( 2 \pi) ^{12} η ^{24} (\tau))

with the third definition implying

j(\tau) can be expressed as a cube, also since 1728{} = 12 ^{3} . \Delta(\tau) = g ^{3} _{2 (\tau)} - 27 g ^{2} _{3 (\tau)} = ( 2 \pi) ^{12} η ^{24} (\tau) , Dedekind eta function η (\tau) , and modular invariants,g _{2(\tau)} = 60 G _{4(\tau)} = 60 \sum _{(m,n)} \left(m + n\tau\right) ^{-4} g _{3(\tau)} = 140 G _{6(\tau)} = 140 \sum _{(m,n)} \left(m + n\tau\right) ^{-6}

where

G _{4(\tau)} , G _{6(\tau)} are Fourier series,\begin{align} G _{4 (\tau)& = (\pi ^{4})/(45)} E _{4(\tau)} \\[ 4 pt] G _{6 (\tau)& = (2 \pi ^{6})/(945)} E _{6(\tau) \end{align}}

and

E _{4(\tau)} , E _{6(\tau)} are Eisenstein series,\begin{align} E _{4(\tau)&=} 1+ 240 \sum ^{inf ty} _{n =1} (n ^{3} q ^{n})/(1- q ^{n}) \\[ 4 pt] E _{6(\tau)&=} 1- 504 \sum ^{inf ty} _{n =1} (n ^{5} q ^{n})/(1- q ^{n}) \end{align}

and

q = e ^{2\pi} (the square of the nome). The -invariant can then be directly expressed in terms of the Eisenstein series as,j(\tau) = 1728 (E ^{3} _{4 (\tau)})/(E ^{3} _{4 (\tau)} - E ^{2} _{6 (\tau)})

with no numerical factor other than 1728. This implies a third way to define the modular discriminant,^[1]

\Delta(\tau) = ( 2 \pi) ^{12} (E ^{3} _{4 (\tau)} - E ^{2} _{6 (\tau)})/(1728)

For example, using the definitions above and

\tau = 2 i , then the Dedekind eta function η ( 2 i) has the exact value,η ( 2 i) = (\Gamma \left( (14 \right))/())/(2 ^{11/8} \pi ^{3/4})

implying the transcendental numbers,

g _{2(2i)} = (11 \Gamma \left( (14 \right))/(^{8}))/(2 ^{8} \pi ^{2}) , g _{3(2i)} = (7 \Gamma \left( (14 \right))/(^{12}))/(2 ^{12} \pi ^{3})

but yielding the algebraic number (in fact, an integer),

j( 2 i) = 1728 (g ^{3} _{2 ( 2 i)})/(g ^{3} _{2 ( 2 i)} - 27 g ^{2} _{3 ( 2 i)}) = 66 ^{3.}

In general, this can be motivated by viewing each as representing an isomorphism class of elliptic curves. Every elliptic curve over is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of . This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by and . This lattice corresponds to the elliptic curve

y ^{2=4x} ^{3- g} ^{2 (\tau)x - g} _{3(\tau)} (see Weierstrass elliptic functions).

Note that is defined everywhere in as the modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots.

The fundamental region

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 holomorphic function invariant under the action of . Quotienting out by its centre

Notes and References

1. Milne. Steven C.. 2000. math/0009130v3. Hankel Determinants of Eisenstein Series. The paper uses a non-equivalent definition of \Delta , but this has been accounted for in this article.