The Modular Invariant

Abstract

We begin with a discussion of an important result in complex analysis called the uniformisation theorem. We have shown how to associate a $\wp$-function with a given lattice L. Thus, $g_{2} = g_{2}(L),g_{3} = g_{3}(L)$ can be viewed as functions on the set of lattices. For a complex number z with imaginary part ℑ(z) > 0, let L _z denote the lattice spanned by z and 1. We will denote the corresponding g ₂, g ₃ associated with L _z by g ₂(z) and g ₃(z). Thus,

$$\displaystyle{g_{2}(z) = 60\sum _{(m,n)\neq (0,0)}(mz + n)^{-4},}$$

and

$$\displaystyle{g_{3}(z) = 140\sum _{(m,n)\neq (0,0)}(mz + n)^{-6}.}$$

We set

$$\displaystyle{\varDelta (z) = g_{2}(z)^{3} - 27g_{ 3}(z)^{2}}$$

which is the discriminant of the cubic defined by the corresponding Weierstrass equation. We first prove: