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 in S the -adic valuation of the order of (R mod p) equals v (m).

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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 (α mod p) is well-defined and has order coprime to ℓ. This set admits a natural density. By refining the method of R. Jones and J. Rouse (2010), 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.

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

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