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
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
![Delta(tau)=(2pi)^(12)[eta(tau)]^(24)](http://mathworld.wolfram.com/images/equations/ModularDiscriminant/NumberedEquation4.gif) |
(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 Unpublished manuscript. 1999.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.
