Reductions of points on algebraic groups, II

TL;DR

This paper extends previous work on the reduction of points on algebraic groups by analyzing the density of primes where a point's order is coprime to a given integer, expressing it through $ ext{l}$-adic integrals and proving its rationality.

Contribution

It generalizes earlier results to arbitrary positive integers, providing a formula for the density and showing it is a rational number with bounded denominator.

Findings

Density expressed as sum of $ ext{l}$-adic integrals

Density is a rational number with bounded denominator

Extends previous prime-specific results to general integers

Abstract

Let $A$ be the product of an abelian variety and a torus over a number field $K$, and let $m$ be a positive integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the reduction $(\alpha \bmod \mathfrak p)$ is well defined and has order coprime to $m$. This set admits a natural density, which we are able to express as a finite sum of products of $\ell$-adic integrals, where $\ell$ varies in the set of prime divisors of $m$. We deduce that the density is a rational number, whose denominator is bounded (up to powers of $m$) in a very strong sense. This extends the results of the paper "Reductions of points on algebraic groups" by Davide Lombardo and the second author, where the case $m$ prime is established.