modular discriminant

Definition 1.

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

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

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

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

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

   $$j(\mathrm{\Lambda })=\frac{g_2^3}{g_2^3-27g_3^2}$$

   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. The $\mathrm{\Delta }$ function (delta function or modular discriminant) is defined to be

   $$\mathrm{\Delta }(\mathrm{\Lambda })=g_2^3-27g_3^2$$

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

   $$\mathrm{\Delta }(\tau )=\mathrm{\Delta }({\mathrm{\Lambda }}_{\tau })={(2\pi i)}^{12}{q}_{\tau }\prod _{n=1}^{\mathrm{\infty }}{(1-q_{\tau }^n)}^{24}={(2\pi i)}^{12}\eta {(\tau )}^{24}$$

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