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.

中文翻译：

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代数群的减分，II

让一种是一个阿贝尔簇和一个数域上的圆环的乘积ķ， 然后让$$m \ge 2$$是一个无平方整数。如果$\alpha \in A(K)$是一个无限阶的点，我们考虑素数的集合$\mathfrak p$的ķ这样的减少$(\alpha \bmod \mathfrak p)$定义明确并且具有互质的顺序米. 这个集合允许一个自然密度，我们可以将其表示为乘积的有限和$\ell$-adic 积分，其中$\ell$在的素因数集合中变化米. 我们推断密度是一个有理数，其分母是有界的（高达米) 在非常强烈的意义上。这扩展了论文的结果代数群上的点约简Davide Lombardo 和第二作者，m prime 的案例成立。