Reductions of points on algebraic groups

TL;DR

This paper investigates the density of primes where the reduction of a point on a product of an abelian variety and a torus has order coprime to a fixed prime, expressing it as an $ extit{ extbf{l}}$-adic integral and proving rationality with bounded denominators.

Contribution

It refines existing methods to express the density as an $ extit{ extbf{l}}$-adic integral without assumptions and establishes the rationality and boundedness of the density's denominator.

Findings

Density can be expressed as an $ extit{ extbf{l}}$-adic integral.

The density is always a rational number.

Denominator of the density is uniformly bounded.

Abstract

Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. 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 $\ell$. This set admits a natural density. By refining the method of R.~Jones and J.~Rouse (2010), we can express the density as an $\ell$-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.