TITLE: Intersection of field extensions of torsion points of non-isogenous elliptic curves
QUESTION [5 upvotes]: Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where $E[p^{\infty}]$ is the set of $p^n$ torsion points of $E$ for all $n$). Then is it true that $k(E[p^{\infty}])\cap k(E'[p^{\infty}]) = k$, or can someone provide a counterexample?

REPLY [6 votes]: By the way, I think that under your hypotheses, your question is really about group theory, not about algebraic geometry.  Namely:  the action of Galois on E[p^infty] x E'[p^infty] gives you a homomorphism
G_K -> SL_2(Z_p) x SL_2(Z_p).
Call the image H.  By your hypothesis, H projects surjectively onto both copies of SL_2(Z_p).  You also know that H is not contained in any conjugate of the diagonal (if it were, E[p^infty] and E'[p^infty] would be isomorphic Galois representations and I'm presuming you're in a situation where Faltings rules that out -- you'd better be, if you want an affirmative answer to your question.)
Now what you have to prove is that a subgroup of SL_2(Z_p) x SL_2(Z_p) which projects surjectively onto each direct summand and which is not conjugate to a subgroup of the diagonal must be finite-index in SL_2(Z_p) x SL_2(Z_p).  This is true for SL_2(F_p) by Hall's lemma and I think you can induct from there (but didn't think about it carefully.)