Geodesic
REDUCTIONS OF POINTS ON ALGEBRAIC GROUPS, II
Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 484-502

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DOI

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. 
BRUIN, PETER; PERUCCA, ANTONELLA. REDUCTIONS OF POINTS ON ALGEBRAIC GROUPS, II. Glasgow mathematical journal, Tome 63 (2021) no. 2, pp. 484-502. doi: 10.1017/S0017089520000336
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     author = {BRUIN, PETER and PERUCCA, ANTONELLA},
     title = {REDUCTIONS {OF} {POINTS} {ON} {ALGEBRAIC} {GROUPS,} {II}},
     journal = {Glasgow mathematical journal},
     pages = {484--502},
     year = {2021},
     volume = {63},
     number = {2},
     doi = {10.1017/S0017089520000336},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000336/}
}
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