On the local-global divisibility of torsion points on elliptic curves and GL2-type varieties

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 GL2-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 numbers p, except a finite number depending only on the isomorphism class of the ring Endk(A), the local-global divisibility by any power of p holds for the torsion points of A.