International Mathematics Research Notices vol:2011 issue:7 pages:293-308
Let G be the product of an abelian variety and a torus deﬁned over a number ﬁeld K . Let R1,...,Rn be points in G(K). Let ℓ be a rational prime, and let a1,...,an be nonnegative integers. Consider the set of primes p of K satisfying the following condition: the ℓ-adic valuation of the order of (Ri mod p) equals ai for every i=1,...,n. We show that this set is either ﬁnite or has a positive natural density. We characterize the n-tuples a1,...,an for which the density is positive. More generally, we study the ℓ-part of the reduction of the points.