Modular Discriminant

Define (cf. the usual nome), where is in the upper half-plane. Then the modular discriminant is defined by

(1)

However, some care is needed as some authors omit the factor of when defining the discriminant (Rankin 1977, p. 196; Berndt 1988, p. 326; Milne 2000).

If and are the elliptic invariants of a Weierstrass elliptic function with periods and , then the discriminant is defined by

(2)

Letting , then

			(3)
			(4)
			(5)

The Fourier series of for , where is the upper half-plane, is

(6)

where is the tau function, and are integers (Apostol 1997, p. 20). The discriminant can also be expressed in terms of the Dedekind eta function by

(7)

(Apostol 1997, p. 51).