  Let G be the product of an abelian variety and a torus defined over a number
field K. Let R be a point in G(K) and let L be a finitely generated subgroup of
G(K). Suppose that for all but finitely many primes p of K the point (R mod p)
belongs to (L mod p). Does it follow that R belongs to L? We answer this
question affirmatively in three cases: if L is cyclic; if L is a free left
End_K G-submodule of G(K); if L has a set of generators (as a group) which is a
basis of a free left End_K G-submodule of G(K). In general we prove that there
exists an integer m (depending only on G, K and the rank of L) such that mR
belongs to the left End_K G-submodule of G(K) generated by L.
