# Adelization for any classical arithmetic subgroup

In the classical setting, we can define automorphic forms on $$\text{SL}_n(\mathbb{R})$$ with respect to any lattice $$\Gamma$$. In fact, for $$n \geq 3$$, all lattices are arithmetic subgroups.

I have encountered the lifting of automorphic forms to the adeles (so to automorphic representations) for $$\Gamma$$ being a congruence subgroup or, more generally, the unit group of an order in a quaternion algebra. I am wondering what are the precise conditions that make this lift possible in general.

Can we "lift" any classical arithmetic subgroup to a compact open subgroup over the adeles? I understand how this might happen when the subgroup is associated to an order in an algebra (the matrix algebra or a division algebra). But there are other constructions of lattices $$\Gamma$$, especially in higher rank. What I am maybe asking is whether simply arithmeticity of the subgroup is enough to make full use of adelic lifts (if they even exist).

For a subgroup to have a meaningful lift to the adeles, it is necessary and sufficient for the subgroup to be a congruence subgroup in the sense that for some $$N$$, the subgroup contains all elements congruent to the identity mod $$N$$.
Given an element of $$SL_n(\mathbb A_{\mathbb Q})$$ (or the same thing for the norm 1 unit group of a central simple algebra)that is integral at every place, it makes sense to consider its congruence class mod $$N$$ (by ignoring all places not dividing $$N$$ and modding out the local ring by $$n$$ at each place dividing $$n$$) and by the Chinese remainder theorem this defines a congruence class in $$SL_n(\mathbb Z/N)$$ or the appropriate analogue. Thus we can define the subgroup of adelic elements that are congruent mod $$N$$ to an element of the group, and this gives an adelic interpretation for automorphic forms.
The property of every finite index subgroup being a congruence group is called the congruence subgroup property, and it is known for non-anisotropic arithmetic groups of real rank $$\geq 2$$, i.e. every arithmetic lattice in $$SL_n(\mathbb R)$$ except for unit groups of division algebras or $$n=2$$. For $$n=2$$ it is known to fail badly and for unit groups of division algebras it is (according to Wikipedia) open.