Let $E/ \Bbb C(t)$ be an elliptic curve over $ \Bbb C(t)$ with nonconstant $j$-invariant $j_E \in \Bbb C(t)-\Bbb C$ and $p>2$ some prime such that it is bigger than an order of a pole $v$ of $j_E$. Assume that the action of $G:={\rm Gal}\bigl(\overline{{\mathbb C}(t)}\big/{\mathbb C}(t)\bigr)$ on the $p$-torsion $E[p]$ is Abelian, ie the image of $G$ in $ \text{Aut}(E[p])={\rm SL}_2({\mathbb Z}/p{\mathbb Z})$ (...as we are in char $0$) is Abelian.
This image can be naturally identified with Galois group $G_p$ of the extension $\Bbb C(t)(E[p])/\Bbb C(t)$ obtained by adjoining all coordinates of $E[p]$ (naturally living in $\overline{{\mathbb C}(t)}$) to $\Bbb C(t)$, right?
Now Here is claimed that under the assumptions above there exist a $p$-cycle $C_p \cong \Bbb Z/p$ in $G_p$ which is ramified at the pole of $j_E$.
(Firstly as "side question" (...which essentially is already posed here), on the phrase that a "$p$-cycle is ramified at a pole", so far I understand this correctly, this ramification of such $p$-cycle $C_p$ at a pole $v$ (regarded as a valuation of global field $\Bbb C(t)$) of $j_E$ phrased in more "down-to-earth" terms means that the field extension $K(C_p)/ \Bbb C(t)$ corresponding to $C_p$ via Galois correspondence ramifies over $v$ ($=$ a pole of $j_E$) in sense of usual algebraic number theory.
Is this exactly what is here meant by "$p$-cycle ramifies at pole of $j_E$"?)
If yes, my central question is - motivated strongly by the concrete problem phrased in the linked question - what do we know in broader sense about the connection of Galois theory of extensions of type $\Bbb C(t)(E[p])/\Bbb C(t)$ ( ie obtained by adding $p$-torsion pts) & and the "information" included in "divisorial data" of $j_E$ (poles & zeroes)? More specifically, my focus lies principally on ramification behavoir of subextensions.
In short, is there any theory tracking relations of zeores/poles of $j_E$ function of elliptic curve over global field with ramification behavior of fields/subfields obtained by adjoining certain torsion points of the curve?
#EDIT: as Chris Wuthrich pointed out this is rather similar to the question linked above I posed before. The difference is that here I'm asking about the "big picture" for the relations between $j$- invariant & ramification data for the Galois theory of torsion fields, which would - in ideal case - give an answer why the "specific" statement I questioned in linked question should hold more less as immediate consequence.
