Transcendental Values of the j-Function

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.