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?



Intersection of field extensions of torsion points of non-isogenous elliptic curves
Since both fields $K(E\_{l^\infty})$ and $K(E'\_{l^\infty})$ contain the $l$-adic cyclotomic extension of $K$, your expectation cannot hold. 
However, this is almost the only obstruction.


In *Propriétés galoisiennes des points d'ordre fini des courbes elliptiques*,
Invent. Math. 15, 259--331 (1972), J-P. Serre 
proved the following Theorem (Theorem 6$''$, p. 325).



> 
> Let $K$ be a number field, let $K^{\rm cycl}$ be the (cyclotomic) extension of $K$
>  generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves
>  such that, over $\bar K$,
> 
> 
> (i) $E$ and $E'$ have no complex multiplication;
> 
> 
> (ii) $E$ and $E'$ are not isogeneous.
> 
> 
> Then, the extensions $K(E\_\infty)$ and $K(E^\prime\_\infty)$ 
>  of $K^{\rm cycl}$ are almost disjoint: 
>  $K(E\_\infty)\cap K(E'\_\infty)$ is finite over $K^{\rm cycl}$.
> 
> 
> 


(By Faltings, hypothesis (ii) is equivalent to the one given by Serre.)


