Modular Discriminant

The modular discriminant is defined, for in the upper half-plane and (cf. the usual nome), 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 half-periods and , then the discriminant is defined by

(2)

Letting be the half-period ratio,

			(3)
			(4)
			(5)

The Fourier series of is

(6)

where the tau function takes integer values (Apostol 1997, p. 20). The discriminant can also be expressed in terms of the Dedekind eta function by

(7)

(Apostol 1997, p. 51).