Year: 2020
Type: article
Abstract: Let A be the product of an abelian variety and a torus over a number field K, and let $$m \ge 2$$ be a square-free 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.
Source: Glasgow Mathematical Journal
Authors: PETER BRUIN, ANTONELLA PERUCCA
Institutions: Leiden University, University of Luxembourg
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,160
OpenAlex ID: W2563949959