As the title suggest, I have a question about abelian (thus Galois) cover of an elliptic curve ramified only above one point. Actually, I'm pretty confused if it exists or not.
To make things clear, I take $E$ an elliptic curve defined over $K$ a number field, and we consider $\pi : C \longrightarrow E$ ramified only above a unique point $P \in E$. We would want $\pi$ galoisian and abelian (i.e the extension of fields corresponding to the covering is galois and abelian).
One one side, if we change $K$ for $\overline{K}$, those covering are classified by the quotient of the fundamental group $\pi_1((E \setminus \{P\})_{\overline{K}})$, and by the description of this group by generators/relations, it corresponds to a group $G$ and elements $x, y, g$ of $G$ such that $[x, y]g = e$, and thus if $G$ is abelian we could obtain only unramified covering. So, it seems that it doesn't exist (over $\overline{K}$) a covering of $E$ galois, abelian, branched only over a single point of $E$.
Working over $K$ and not $\overline{K}$, it's a little bit different cause now the galois covering are given by the quotient of $\pi_1((E \setminus \{P\}))$, and we have : $$ 0 \rightarrow \pi_1((E \setminus \{P\})_{\overline{K}}) \rightarrow \pi_1((E \setminus \{P\})) \rightarrow Gal(\overline{K}/K) \rightarrow 0$$
On the other side, suppose we want $\pi$ cyclic of degree $l$, $l$ prime (recall that $\text{char}(K)=0$). Then, if $K$ contains a primitive $l$-th root of unity, by the Kummer theory it corresponds to an extension $K(E)(f^{1/l})/K(E)$ where $f \in K(E)^\times/K(E)^{\times l}$, and the ramification point can be only those which are the poles/zeros of $f$, and are exactly thoses zeros/poles $P$ such that $\text{ord}_P(f)$ is prime to $l$. So, it seems to exist such a galoisian and abelian covering $\pi$ ramified only above one point of $E$, no ?
What I didn't understand ?
Thank you !