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).

REFERENCES:

Apostol, T. M. "The Discriminant " and "The Fourier Expansions of  and ." §1.11 and 1.15 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 14 and 20-22, 1997.

Berndt, B. C. Ramanujan's Notebooks, Part II. New York: Springer-Verlag, p. 326, 1988.

Brezhnev, Y. V. "Uniformisation: On the Burnside Curve ." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.

Milne, S. C. "Hankel Determinants of Eisenstein Series." 13 Sep 2000. http://arxiv.org/abs/math.NT/0009130.

Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.

Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, p. 196, 1977.