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).