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 etafunction 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 theWeierstrass eta function, $\eta (w;\mathrm{\Lambda })$.

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

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

   where $g_2$ and $g_3$ are certain multiples of the Eisensteinseries of weight $4$ and $6$ (see http://planetmath.org/encyclopedia/ExamplesOfEllipticFunctions.htmlthisentry).

3. The $\mathrm{\Delta }$ function (delta function ormodular 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

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