# 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.

## Key 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.