Modular Discriminant
Define q=e^(2piitau) (cf. the usual nome), where tau is in the upper half-plane. Then the modular discriminant is defined by
However, some care is needed as some authors omit the factor of (2pi)^(12) when defining the discriminant (Rankin 1977, p. 196; Berndt 1988, p. 326; Milne 2000).
If g_2(omega_1,omega_2) and g_3(omega_1,omega_2) are the elliptic invariants of a Weierstrass elliptic function P(z|omega_1,omega_2)=P(z;g_2,g_3) with periods omega_1 and omega_2, then the discriminant is defined by
| Delta(omega_1,omega_2)=g_2^3-27g_3^2. |
(2)
|
Letting tau=omega_2/omega_1, then
The Fourier series of Delta(tau) for tau in H, where H is the upper half-plane, is
where tau(n) is the tau function, and tau(n) are integers (Apostol 1997, p. 20). The discriminant can also be expressed in terms of the Dedekind eta function eta(tau) by
| Delta(tau)=(2pi)^(12)[eta(tau)]^(24) |
(7)
|
(Apostol 1997, p. 51).
See also
Dedekind Eta Function, Elliptic Invariants, Klein's Absolute Invariant, Nome, Tau Function, Weierstrass Elliptic FunctionExplore with Wolfram|Alpha
References
Apostol, T. M. "The Discriminant Delta" and "The Fourier Expansions of Delta(tau) and J(tau)." §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 y^2=x^5-x." 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 Unpublished manuscript. 1999.Rankin, R. A. Modular Forms and Functions.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.Referenced on Wolfram|Alpha
Modular DiscriminantCite this as:
Weisstein, Eric W. "Modular Discriminant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModularDiscriminant.html