Journal of Number Theory vol:129 issue:2 pages:469-476
Let G be the product of an abelian variety and a torus deﬁned over a number ﬁeld K. Let R be a K-rational point on G of inﬁnite order. Call n_R the number of connected components of the smallest algebraic K -subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but ﬁnitely many primes p of K . Furthermore, let m > 0 be
a multiple of n R and let S be a ﬁnite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every ℓ in S the ℓ-adic valuation of the order of (R mod p) equals v_ℓ(m).