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

TL;DR

This paper establishes a link between the order of a rational point on an abelian variety times a torus over a number field and its reductions modulo primes, providing a way to predict valuations of these reductions.

Contribution

It introduces a method to determine the divisibility properties of the order of reductions of rational points on products of abelian varieties and tori, connecting algebraic and arithmetic properties.

Findings

n_R divides the order of (R mod p) for almost all primes p

Existence of primes with prescribed l-adic valuations of the order

Characterization of the divisibility of reduction orders in terms of algebraic subgroups

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).