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).

arXiv commentFinal version. To appear on Journal of Number Theory

References in corpus (1)

• Galois theory of iterated endomorphisms

Cited by in corpus (9)

• Galois theory of iterated endomorphisms
• Finite dimensional Hopf actions on algebraic quantizations
• Two variants of the support problem for products of abelian varieties and tori
• Reductions of points on algebraic groups
• Reductions of points on algebraic groups, II
• Reductions of subgroups of the multiplicative group
• The constant of the support problem for abelian varieties
• On the reduction of points on abelian varieties and tori
• Characterizing Abelian Varieties by the Reductions of the Mordell-Weil Group