I'd like to understand the fields of definition of abelian covers of elliptic curves In particular, let $E$ be an elliptic curve over a field $K$ (say, a number field).
I'd like to understand the possible abelian Galois covers $X\rightarrow E$ (ie, in terms of the standard 'arithmetic' of $E$) - in particular, the ones which are ramified. If we fix a branch divisor $D\subset E$ (ie, require that the map can only ramify over $D$), then the unramified abelian covers of $E-D$ are classified by the abelianized fundamental group $\pi_1(E-D)^{ab}$, or equivalently the geometric fundamental group $\Pi_D := \pi_1((E-D)_{\overline{K}})^{ab}$ equipped with an action of the absolute galois group $G_K$. If $D_{\overline{K}} = \{x_1,\ldots,x_r\}$, then $\Pi_D$ should be isomorphic to $\widehat{\mathbb{Z}}^{2+r-1}$, and is an extension of the usual Tate module $\prod_\ell T_\ell(E)\cong \pi_1(E_{\overline{K}})$ by $K := \widehat{\mathbb{Z}}^{r-1}$ on which $G_K$ acts via the cyclotomic character. Ie, we should have an exact sequence of free pro-abelian $G_K$-groups $$0\rightarrow K\rightarrow\Pi_D\rightarrow\pi_1(E_{\overline{K}})\rightarrow 0$$
(A) Is the above sequence split exact as $G_K$-groups? (of course it is split as abstract groups) Is it split when $D$ is torsion (ie, $D\subset E[m]$ for some $m$).
(B) Is it possible to describe the minimum field over which all abelian covers of $E_{\overline{K}}$ which have torsion branch points are defined? I think I can prove that this field is $K(E^{tors})$, though I'm not sure if there is an easier proof.
