Loosely speaking, are elliptic Kummer extensions big? More concretely:


Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and
let $F$ be a subfield of $\overline{\mathbb{Q}}$ containing the
coordinates of all the $p$-power torsion of $E$. Given $c > 0$, does
there
exist $N > 0$ (depending only on $E$, $F$, and $p$) such that:
If P is a point of $E(\overline{\mathbb{Q}})$ whose image in
$(E(\overline{\mathbb{Q}})/E(F))$ has
order $p^n$ (in other words, $[p^n]P \in E(F)$, but this does not hold for
any smaller power of $p$) with $n \geq N$, then $[F(P):F] \geq c$?


Remarks:


* I am aware of some Theorems of Basmakov, Ribet, and Bertrand that are roughly in this direction, and in fact more general cases than elliptic curves are considered (e.g. Ribet's paper "Kummer theory on extensions of abelian varieties by tori" -- Duke Math J. 1979). One shortcoming of these excellent papers is that the results are only for almost every prime. I am very interested in the prime 2 (unfortunately!).
* $\operatorname{Gal}(F(P)/F)$ is naturally a subgroup of $E[p^n]$.
* The G\_m analogue is easy -- the degree of a Kummer extension is as
big as it could be.
* If it's easier, the most important case for me is when $E$
has CM and $F$ is a finite extension of $\mathbb{Q}(E\_{tors})$.
* Does the following make sense: Given a point $Q \in E(F)$, there is a
representation $G\_F \to (\mathbb{Z}\_p)^2$ and more generally $G\_\mathbb{Q}
\to (\mathbb{Z}\_p)^2 \rtimes \operatorname{GL}\_2(\mathbb{Z\_p})$ coming
from the Galois action on points $P$ satisfying $[p^n]P = Q, n \in
\mathbb{N}$, and in this context what I'm asking is in the spirit of
Serre's Theorem on the large image of the $p$-adic representation, but I'm asking the image to be large *uniformly in $Q$?*



Are elliptic Kummer extensions big?
$Gal(F(P)/F)$ is indeed a subgroup of $E[p^{n}]$. We see that it is not contained in $E[p^{n-1}]$, because otherwise $p^{n-1}P\in E(F)$. Thus, since $E[p^{n-1}]$ consists of all the elements of order less than $p^n$, order of $Gal(F(P)/F)$ is at least $p^N$. So we may take any $N \leq \log\_p c$


