Modular Discriminant (original) (raw)

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Define q=e^(2piitau) (cf. the usual nome), where tau is in the upper half-plane. Then the modular discriminant is defined by

Delta(tau)=(2pi)^(12)qproduct_(r=1)^infty(1-q^r)^(24). (1)

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

Delta(tau)=(2pi)^(12)sum_(n=1)^inftytau(n)e^(2piintau), (6)

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 Function

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References

Apostol, T. M. "The Discriminant Delta" and "The Fourier Expansions of Delta(tau) and J(tau)." ยง1.11 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 Unpublished manuscript. 1999.Rankin, R. A. Modular Forms and Functions.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.

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Modular Discriminant

Cite this as:

Weisstein, Eric W. "Modular Discriminant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModularDiscriminant.html

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