Reductions of points on algebraic groups

Let A be the product of an abelian variety and a torus defined over a number field K. Fix some prime number ℓ. If α∈A(K) is a point of infinite order, we consider the set of primes p of K such that the reduction (αmodp) is well-defined and has order coprime to ℓ. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3)100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an ℓ-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of ℓ) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.