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modular discriminant (Definition)
Definition 1   Let $ \Lambda\subset\mathbb{C}$ be a lattice.
  1. Let $ q_{\tau}=e^{2\pi i \tau}$. The Dedekind eta function is defined to be
    $\displaystyle \eta(\tau)=q_{\tau}^{1/24}\prod_{n=1}^{\infty}(1-q_{\tau}^n)$
    The Dedekind eta function should not be confused with the Weierstrass eta function, $ \eta(w;\Lambda)$.
  2. The $ j$-invariant, as a function of lattices, is defined to be:
    $\displaystyle j(\Lambda)=\frac{g_2^3}{g_2^3-27g_3^2}$
    where $ g_2$ and $ g_3$ are certain multiples of the Eisenstein series of weight $ 4$ and $ 6$ (see this entry).
  3. The $ \Delta$ function (delta function or modular discriminant) is defined to be
    $\displaystyle \Delta(\Lambda)=g_2^3-27g_3^2$
    Let $ \Lambda_{\tau}$ be the lattice generated by $ 1,\tau$. The $ \Delta$ function for $ \Lambda_{\tau}$ has a product expansion
    $\displaystyle \Delta(\tau)=\Delta(\Lambda_{\tau})=(2\pi i)^{12}q_{\tau}\prod_{n=1}^{\infty}(1-q_{\tau}^n)^{24}=(2\pi i)^{12}\eta(\tau)^{24}$



"modular discriminant" is owned by alozano.
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See Also: elliptic function, j-invariant, Weierstrass sigma function, discriminant, discriminant, Ramanujan tau function

Other names:  delta function
Also defines:  modular discriminant, Dedekind eta function
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Cross-references: product, generated by, weight, Eisenstein series, multiples, function, Weierstrass eta function, lattice
There are 8 references to this entry.

This is version 3 of modular discriminant, born on 2003-08-25, modified 2008-02-26.
Object id is 4651, canonical name is ModularDiscriminant.
Accessed 6750 times total.

Classification:
AMS MSC: 33E05 (Special functions :: Other special functions :: Elliptic functions and integrals)
Pending Errata and Addenda
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