# modular discriminant

###### Definition 1.

Let $\Lambda\subset\mathbb{C}$ be a lattice.

1. 1.

Let $q_{\tau}=e^{2\pi i\tau}$. The Dedekind eta function is defined to be

 $\eta(\tau)=q_{\tau}^{1/24}\prod_{n=1}^{\infty}(1-q_{\tau}^{n})$

The Dedekind eta function should not be confused with the Weierstrass eta function, $\eta(w;\Lambda)$.

2. 2.

The $j$-invariant, as a function of lattices, is defined to be:

 $j(\Lambda)=\frac{g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}$

where $g_{2}$ and $g_{3}$ are certain multiples of the Eisenstein series of weight $4$ and $6$ (see http://planetmath.org/encyclopedia/ExamplesOfEllipticFunctions.htmlthis entry).

3. 3.

The $\Delta$ function (delta function or modular discriminant) is defined to be

 $\Delta(\Lambda)=g_{2}^{3}-27g_{3}^{2}$

Let $\Lambda_{\tau}$ be the lattice generated by $1,\tau$. The $\Delta$ function for $\Lambda_{\tau}$ has a product expansion

 $\Delta(\tau)=\Delta(\Lambda_{\tau})=(2\pi i)^{12}q_{\tau}\prod_{n=1}^{\infty}(% 1-q_{\tau}^{n})^{24}=(2\pi i)^{12}\eta(\tau)^{24}$
 Title modular discriminant Canonical name ModularDiscriminant Date of creation 2013-03-22 13:54:09 Last modified on 2013-03-22 13:54:09 Owner alozano (2414) Last modified by alozano (2414) Numerical id 6 Author alozano (2414) Entry type Definition Classification msc 33E05 Synonym delta function Related topic EllipticFunction Related topic JInvariant Related topic WeierstrassSigmaFunction Related topic Discriminant Related topic DiscriminantOfANumberField Related topic RamanujanTauFunction Defines modular discriminant Defines Dedekind eta function