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Concept
J-invariant
Summary
In mathematics, Felix 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).
The j-invariant can be defined as a function on the upper half-plane H = {τ ∈ C, Im(τ) > 0},
with the third definition implying can be expressed as a cube, also since 1728.
The given functions are the modular discriminant , Dedekind eta function , and modular invariants,
where , are Fourier series,
and , are Eisenstein series,
and (the square of the nome). The j-invariant can then be directly expressed in terms of the Eisenstein series as,
with no numerical factor other than 1728. This implies a third way to define the modular discriminant,
For example, using the definitions above and , then the Dedekind eta function has the exact value,
implying the transcendental numbers,
but yielding the algebraic number (in fact, an integer),
In general, this can be motivated by viewing each τ as representing an isomorphism class of elliptic curves. Every elliptic curve E over C is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of C. This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by 1 and τ ∈ H. This lattice corresponds to the elliptic curve (see Weierstrass elliptic functions).
Note that j is defined everywhere in H as the modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots.
It can be shown that Δ is a modular form of weight twelve, and g2 one of weight four, so that its third power is also of weight twelve.
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