Prescribing valuations of the order of a point in the reductions of abelian varieties and tori

Abstract

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 ) .