Division Field of a nonCM elliptic curve

Question

This might be a ridiculous question, but please bear with me.

Let $E$ be an elliptic curve over a $p$-adic field $K$. Denote by $K(E_{p^∞}):=\bigcup_{n∈Z≥1} K(E[p^n])$ the field extension obtained by adjoining to $K$ the coordinates of all $p$-power torsion points of $E$. By the Weil pairing, we know that all the $p$-power roots of unity lie in $K(E_{p^∞})$.

If $E$ has multiplicative reduction over $K$, then we can describe $K(E_{p^∞})$ very well by the theory of Tate curves. It is known that there is a finite extension $F$ over $K$, of degree at most $2$, and a unique element $α∈F^×$ with negative valuation such that $$K(E_{p^∞})=F(μ_{p^∞},α^{p^{−∞}})= \bigcup_{m∈\mathbb{Z}≥1}K(μ_{p^∞},α^{p^{−m}})$$, where $μ_{p^∞}$ denotes the group of all $p$-power roots of unity.

Now, consider the case when $E$ has good ordinary reduction without complex multiplication. I was wondering if something of the same sort as in the above description holds. More precisely:

Question: Is it possible that there is an element, say $β$, of $K(E_{p^∞})$, which is not a root of unity, such that all the $p$-power roots of $β$ lie in $K(E_{p^∞})$?

I could ask the same thing for other cases on $E$ but I am most interested in the above non-CM case.

Answer 1

$E[p^n]$ is, over the maximal unramified extension of $K$ (obtained by adding all prime to $p$ roots of unity), an extension of $\mathbb{Z}/p^n$ by $\mu_{p^n}$. The extension class is a principal unit, called the Serre-Tate parameter. To trivialize the extension you need to take the $p^n$-th root of this parameter. The $\mu_{p^n}$ is, of course, trivialized by adjoining the $p$-power roots of unity. So the answer to your question is yes, with $\beta$ being the Serre-Tate parameter.