On the reduction of points on abelian varieties and tori

TL;DR

This paper investigates the distribution of primes where the reductions of points on a product of an abelian variety and a torus have specified l-adic valuations, establishing conditions for positive density.

Contribution

It characterizes when the set of primes with prescribed reduction properties has positive density for points on products of abelian varieties and tori.

Findings

The set of primes with specified l-adic valuations has a natural density.

Conditions are provided for when this density is positive.

The study extends to the l-part of the reduction of points.

Abstract

Let G be the product of an abelian variety and a torus defined over a number field K. Let R_1,..., R_n be points in G(K). Let l be a rational prime and let a_1,..., a_n be non-negative integers. Consider the set of primes p of K satisfying the following condition: the l-adic valuation of the order of (R_i mod p) equals a_i for every i=1,...,n. We show that this set has a natural density and we characterize the n-tuples a_1,..., a_n for which the density is positive. More generally, we study the l-part of the reduction of the points.