Two variants of the support problem for products of abelian varieties and tori

Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of divides the order of . Then there exist a K-endomorphism ϕ of G and a non-zero integer c such that ϕ(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the ℓ-adic valuation of the order for some fixed rational prime ℓ (ℓ-adic support problem).