Reductions of points on algebraic groups, II

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 $(\alpha \mathrm{mod}\mathfrak{p})$ 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.

References in corpus (3)

• Computing images of Galois representations attached to elliptic curves
• Prescribing valuations of the order of a point in the reductions of abelian varieties and tori
• Reductions of points on algebraic groups