J-invariant

J function or absolute invariant plays an important role in the theory of elliptic functions, and modular forms.

Definition

For is

Here is the discriminant; and are the Eisenstein series to the grid.

Properties

The j- function is holomorphic on, the term absolute invariant is due to the transformation behavior under the substitutions of the modular group, it is namely:

The j- function is surjective onto. Then, for points and only if there is a complex number, which leads the grid on the grid, so if and only if the quotient and the elliptical curves are isomorphic.

Fourier expansion

The j- function can be used in a Fourier series development:

With

All Fourier coefficients:

Are natural numbers. For their growth is considered, the asymptotic formula

Was the 1932 by Petersson and it proved independently in 1938 by Rademacher.

The Fourier coefficients are linear combinations of the dimensions of irreducible representations of the Monster group with small integer coefficients. This follows from a deep mathematical relationship that has been suggested by McKay, Conway, Norton and proved by Richard Borcherds ( " monstrous moonshine ").