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REDUCTIONS OF POINTS ON ALGEBRAIC GROUPS, II

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry Algebraic groups

Published online by Cambridge University Press:  28 July 2020

PETER BRUIN  and
ANTONELLA PERUCCA
PETER BRUIN
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands, e-mail: [email protected]
ANTONELLA PERUCCA
Affiliation:
Department of Mathematics, University of Luxembourg, 6, Avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg, e-mail: [email protected]
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Abstract

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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.

MSC classification

Primary: 11F80: Galois representations
Secondary: 14L10: Group varieties 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 484 - 502
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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