Title: On Almost Strong Approximation in Reductive Algebraic Groups Show Chinese title

Title: 关于半单代数群中的几乎强逼近

Abstract: We investigate a slight weakening of the classical property of strong approximation, which we call almost strong approximation, for connected reductive algebraic group over global fields with respect to special sets of valuations. While nonsimply connected groups (in particular, all algebraic tori) always fail to have strong approximation - and even almost strong approximation -- with respect to any finite set of valuations, we show that under appropriate assumptions they do have almost strong approximation with respect to (infinite) tractable sets of valuations, i.e. those sets that contain all archimedean valuations and a generalized arithmetic progression minus a set having Dirichlet density zero. Almost strong approximation is likely to have a variety of applications, and as an example we use almost strong approximation for tori to extend the essential part of the result of Radhika and Raghunathan on the congruence subgroup problem for inner forms of type $\textsf{A}_n$ to all absolutely almost simple simply connected groups. Show Chinese abstract

Abstract: 我们研究了经典强逼近性质的一个微弱形式，我们称之为几乎强逼近，针对全局域上的连通约化代数群，在特定的赋值集合上。 虽然非单连通群（特别是所有代数环）在任何有限的赋值集合上总是无法满足强逼近——甚至几乎强逼近——但我们证明，在适当的假设下，它们在（无限）可处理的赋值集合上具有几乎强逼近，即那些包含所有阿基米德赋值和一个广义算术级数减去一个狄利克雷密度为零的集合的集合。 几乎强逼近可能有各种应用，作为例子，我们使用环的几乎强逼近来扩展Radhika和Raghunathan关于类型$\textsf{A}_n$的内形式的同余子群问题的结果的核心部分，以适用于所有绝对几乎简单的单连通群。