Prove that $E(\mathbb{C})^{\text{tor}} \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z}$.

Question

Let $E$ be an elliptic curve over $\mathbb{C}$. We know that $E(\mathbb{C}) \cong \mathbb{C}/L$ (this is a group isomorphism) for some lattice $L \subset \mathbb{C}$. Using this fact prove that

$$\begin{aligned} E(\mathbb{C})^{\text{tor}} \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z} \end{aligned}$$

Is $E(\mathbb{C})^{\text{tor}}$ a finitely generated abelian group?

I know that to show that those groups are isomorphic, we need to simply construct the mapping of those groups that is the isomorphism. Some other thoughts that I have is that since $\mathbb{Q}/\mathbb{Z}$ is torsion, we have $\mathbb{Q}/\mathbb{Z} = (\mathbb{Q}/\mathbb{Z})^{\text{tor}}$, and that working backward i.e. proving that $\mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z} \simeq E(\mathbb{C})^{\text{tor}}$ might work. However, I am stuck because I was not able to find the way to use the fact (that $E(\mathbb{C}) \simeq \mathbb{C}/L$) to prove these isomorphic groups.

Any suggestions or ideas?

Answer 1

Yes, $E(\Bbb C)\cong\Bbb C/L$ for a lattice $L$. And $\Bbb C/L$ is isomorphic to...?

(You may be aware that $\Bbb Q/\Bbb Z$ is the torsion subgroup of ____)

$\Bbb C/L\cong (\Bbb R\oplus\Bbb R)/(\Bbb Z\oplus\Bbb Z)\cong(\Bbb R/\Bbb Z)\oplus(\Bbb R/\Bbb Z$), the blank is referring to $\Bbb R/\Bbb Z$