Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. If $\unicode[STIX]{x1D6FC}\in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak{p}$ of $K$ such that the reduction $(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$ is well-defined and has order coprime to $\ell$. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3)100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an $\ell$-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.

中文翻译：

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代数群上的减分

让$澳元是在数域上定义的阿贝尔簇和圆环的乘积$K$. 修复一些素数$\ell$. 如果$\unicode[STIX]{x1D6FC}\in A(K)$是一个无限阶的点，我们考虑素数的集合$\mathfrak{p}$的$K$这样的减少$(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$是明确定义的并且具有互质的顺序$\ell$. 这组承认自然密度。通过改进 Jones 和 Rouse [Galois 迭代自同态理论的方法，过程。伦敦。数学。社会党。(3)100(3) (2010), 763–794。Jeffrey D. Achter 的附录 A]，我们可以将密度表示为$\ell$-adic 积分，无需任何假设。我们还证明了密度总是一个有理数，其分母（高达$\ell$) 在非常强烈的意义上是一致有界的。对于椭圆曲线，我们描述了一种计算密度的策略，它涵盖了所有可能的情况。