torsion points on complex torus

Question

I’m not sure how to convince myself that "set of torsion points is dense on complex torus".

Is this obvious? I could not find a reference or a proof.

Could anyone help me clarify or verify this?

Answer 1

Let $E = \mathbb{C}/\Lambda$ be an elliptic curve over $\mathbb{C}$ (i.e., a complex torus), where $\Lambda = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2$ is a lattice (and the $\omega$'s are linearly independent over $\mathbb{R}$). We want to show that for any nonempty open ball $U$ contains a torsion point $p$.

We have the following claim: the (image under the quotient map of) $p = (m_1\omega_1 + m_2 \omega_2) / n$ is torsion for each $n, m_1, m_2 \in \mathbb{Z}$. But this is easy to see since $np \in \Lambda$.

Well, now let $U' \subset \mathbb{C}$ be the inverse image of the ball $U$ under the quotient map $\mathbb{C} \to \mathbb{C}/\Lambda$. Note that $U'$ is nonempty since $U$ is. The result follows from the claim that $U'$ contains a $p$ as above.

To see this notice that the $\mathbb{Q}$-span of $\{\omega_1, \omega_2\}$ is dense in $\mathbb{C}$, so has nonempty intersection with $U'$ - say containing $p = (m_1/n_1) \omega_1 + (m_2/n_2)\omega_2$. Thus (the image of) $p = (m_1n_2\omega_1 + m_2n_1 \omega_2)/(n_1n_2)$ is a torsion point, contained in $U$.