I'm trying to understand the first part of the proof of Lemma 2 in https://www.emis.de/journals/BAG/vol.46/no.2/b46h2hei.pdf.
Given $r > 0$, they claim that for an elliptic curve $E$ there exists another elliptic curve $\widetilde{E}$ with $\pi_r: \widetilde{E} \rightarrow E$ such that $E = \widetilde{E}/G$ where $G = \mathbb{Z}/r$.
They provide a construction as follows: take $M$ to be a line bundle of order $r$, and let $\widetilde{E}= \mathcal{Spec}(\mathscr{O}_E \oplus M \oplus \cdots \oplus M^{\otimes {r-1}})$.
My first question is why does such a line bundle exist (can I always take the $r$th root of a line bundle on an elliptic curve?), and why does $\widetilde{E}$ have the desired properties? (I don't even understand why it ends up being an elliptic curve).
I understand that over $\Bbb C$, I can just take the $r$-fold covering space of the complex torus, but I'm looking for an answer that works more generally.
Thanks.