REDUCTIONS OF POINTS ON ALGEBRAIC GROUPS, II

Glasgow Mathematical Journal, vol. 63, pp. 484–502

Abstract

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.

Authors 2

1. Peter Bruin Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands, e-mail:

2. Antonella Perucca

   University of Luxembourg

   Affiliation as printed

   Department of Mathematics, University of Luxembourg, 6, Avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg, e-mail: