Weierstrass's elliptic functions#Modular discriminant

The discriminant Δ is ​​a holomorphic function on the upper half-plane.

It plays an important role in the theory of elliptic functions and modular forms.

Definition

For his,

There are and the Eisenstein series to the grid.

Product Development

The discriminant Δ can be divided into an infinite product development, we have:

From the product representation follows immediately that has no zeros.

The discriminant Δ is ​​closely related to the Dedekind η - function, it is.

Transformation behavior

The discriminant Δ is ​​an entire modular form of weight 12, ie under the substitutions of

Applies:

The discriminant Δ has in a zero and is the simplest example of a so-called top form (English cusp form).

Fourier expansion

The discriminant Δ can be divided into a Fourier series development:

The Fourier coefficients are all integers and are called Ramanujan tau - function

Referred to. This is a multiplicative number-theoretic function, ie

As evidenced in 1920 by Louis Mordell. More precisely, the formula

For the first values ​​of the tau - function is:

To date, not a " simple" arithmetic definition of the tau - function is known. Likewise, still unknown whether the condition laid down by DH Lehmer conjecture

Ramanujan conjectured that goes for primes p:

This conjecture was proved in 1974 by Deligne.

The satisfy the congruence of Ramanujan discovered already

With