On the congruence subgroup problem for anisotropic groups of inner type $$A_n$$

Abstract

Let G be an absolutely almost simple simply connected algebraic group of inner form of type $$A_n$$ defined and anisotropic over a number field k. Let V denote a complete set of inequivalent valuations of k and $$V_f$$ (resp. $$\infty $$) the subset of non-archimedean (resp. archimedean) valuations in V. This work deals with the congruence subgroup problem for groups G of inner form of type $$A_n$$ with respect to a particular class of valuations $$S\subset V$$ which we term as d-amenable. Fix a realization $$G \subset {\mathrm {GL}}(n)$$ as a k-subgroup and set $$\Lambda _S = G(k)\cap GL(n, {\mathcal {O}}_S)$$ where $${\mathcal {O}}_S$$ is the ring of S-integers in k. Then we show that if S is d-amenable and $$\Gamma $$ is any subgroup of finite index in $$\Lambda _S$$, there exists a finite set $$S(\Gamma )$$ and positive integers $$\{r_v \mid v \in S(\Gamma )\}$$ such that $$\Gamma $$ contains $$\{\gamma \in \Lambda _S\mid \gamma \in (1 +{\mathfrak {p}}_v^{r_v})\ \text {for}\ v\in S(\Gamma )\}$$. The d-amenability property holds for any S with $$V\smallsetminus S$$ finite. We also show that the set of primes in V in an arithmetic progression is d-amenable for any d so that we recover the result of Prasad and Rapinchuk (Proc Steklov Inst Math 292(1):216–246, 2016).