On Almost Strong Approximation in Reductive Algebraic Groups
Andrei S. Rapinchuk, Wojciech Tralle
TL;DR
This work introduces and develops almost strong approximation for connected reductive groups over global fields, extending classical strong approximation which fails for nonsimply connected groups such as tori. By analyzing the $S$-adelic closures of $K$-rational points and leveraging Dirichlet density and Chebotarev-type density, the authors prove finiteness of the index $[G(\mathbb{A}_{K,S}) : \overline{G(K)}^{(S)}]$ under precise conditions, first for tori (and then for general reductive groups) and for tractable sets of valuations. The key technical toolkit includes class field theory, Nakayama–Tate duality, and bounded cohomology ($H^1$), together with the construction of special covers to reduce the reductive case to torus and simply connected semisimple cases. As applications, the paper extends results on the congruence subgroup problem to inner forms of type $\textsf{A}_n$ and, more generally, to absolutely almost simple simply connected groups, establishing finiteness or triviality of the congruence kernel in broad tractable settings. The work also clarifies obstructions to almost strong approximation by exhibiting a counter-example where density conditions fail, and it integrates recent refinements via Dirichlet density to broaden the scope of almost strong approximation beyond arithmetic progressions.
Abstract
We investigate a slight weakening of the classical property of strong approximation, which we call almost strong approximation, for connected reductive algebraic groups 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 certain infinite sets of valuations that can be characterized in terms of Dirichlet density and include tractable sets of valuations, i.e. those sets that contain all archimedean valuations and a generalized arithmetic progression minus a set of Dirichlet density zero. Almost strong approximation is likely to have a variety of applications, and as an example we use almost strong approximation in tori with respect to tractable sets 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. This version of the paper has been updated to reflect recent results of Y. Cao and Y. Wang (arXiv:2511.00824).