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 this 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

   $$\Delta(\tau)=\Delta(\Lambda_{{\tau}})=(2\pi i)^{{12}}q_{{\tau}}\prod_{{n=1}}^{{\infty}}(1-q_{{\tau}}^{n})^{{24}}=(2\pi i)^{{12}}\eta(\tau)^{{24}}$$