Transcendental Values of the j-Function

Abstract

Let L and M be two lattices with corresponding Weierstrass functions $\wp$ and $\wp ^{{\ast}}$. We begin by showing that if $\wp$ and $\wp ^{{\ast}}$ are algebraically dependent, then there is a natural number m such mM⊆ L. Indeed suppose that $\wp$ and $\wp ^{{\ast}}$ are as above and there is a polynomial $P(x,y) \in \mathbb{C}[x,y]$ such that $P(\wp,\wp ^{{\ast}}) = 0$. Then for some rational functions a _i (x) and some natural number n, we have

$$\displaystyle{\wp (z)^{n} + a_{ n-1}(\wp ^{{\ast}}(z))\wp (z)^{n-1} + \cdots + a_{ 0}(\wp ^{{\ast}}(z)) = 0.}$$

Choose $z_{0} \in \mathbb{C}$ so that $\wp ^{{\ast}}(z_{0})$ is not a pole of the a _i (z) for 0 ≤ i ≤ n − 1. This can be done since the a _i (z) are rational functions and so there are only finitely many values to avoid in a fundamental domain. Then

$$\displaystyle{\wp (z_{0})^{n} + a_{ n-1}(\wp ^{{\ast}}(z_{ 0}))\wp (z_{0})^{n-1} + \cdots + a_{ 0}(\wp ^{{\ast}}(z_{ 0})) = 0.}$$

If ω ^∗ ∈ M, then we get

$$\displaystyle{\wp (z_{0} +\omega ^{{\ast}})^{n} + a_{ n-1}(\wp ^{{\ast}}(z_{ 0}))\wp (z_{0} +\omega ^{{\ast}})^{n-1} + \cdots + a_{ 0}(\wp ^{{\ast}}(z_{ 0})) = 0.}$$

Thus $\wp (z_{0} +\omega ^{{\ast}})$, as ω ^∗ ranges over elements of M, are also zeros of the polynomial

$$\displaystyle{z^{n} + a_{ n-1}(\wp ^{{\ast}}(z_{ 0}))z^{n-1} + \cdots + a_{ 0}(\wp ^{{\ast}}(z_{ 0})) = 0.}$$

In particular, this is true of multiples of ω ₁ ^∗ and ω ₂ ^∗. We therefore get infinitely many roots of the above polynomial equation unless mM ⊆ L for some positive natural number m. We record these observations in the following.