  Let $p$ be a prime number and let $k$ be a number field. Let $E$ be an
elliptic curve defined over $k$. We prove that if $p$ is odd, then the
local-global divisibility by any power of $p$ holds for the torsion points of
$E$. We also show with an example that the hypothesis over $p$ is necessary.
  We get a weak generalization of the result on elliptic curves to the larger
family of ${\rm GL}_2$-type varieties over $k$. In the special case of the
abelian surfaces $A/k$ with quaternionic multiplication over $k$ we obtain that
for all prime $p$, except a finite number depending on $A$, the local-global
divisibility by any power of $p$ holds for the torsion points of $A$
