The constant of the support problem for abelian varieties

Let A be an abelian variety defined over a number field K and let P and Q   be points in A(K)A(K) satisfying the following condition: for all but finitely many primes pp of K  , the order of (Qmodp) divides the order of (Pmodp). Larsen proved that there exists a positive integer c such that cQ   is in the EndK(A)EndK(A)-module generated by P. We study the minimal value of c and construct some refined counterexamples.