Let $K$ be a global field and set $O := \prod_{v\nmid \infty} O_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}_n(O) = \prod_v \mathrm{GL}_n(O_v)$ with the product of the $v$-adic topologies and suppose that $H \le \mathrm{GL}_n(O)$ is a subgroup of finite index.

- Is $H$ necessarily open?
- Is $H$ necessarily a congruence subgroup? (I.e., does it contain $\mathrm{Ker}(\mathrm{GL}_n(O) \rightarrow \mathrm{GL}_n(O/NO))$ for some integer $N$?)

A positive answer to 2. would imply a positive answer to 1.

I will accept a complete answer to either 1. or 2. as an answer. Also, I would be especially happy if you discussed a generalization of 1. and 2. for arbitrary reductive group schemes (or even a more general class of group schemes) over the ring of integers of $K$.

knowis finite index but for which you can't see the answers to either #1 or #2 immediately? That is: does this question have some motivation, or is it just idle curiosity? $\endgroup$ – user27920 Sep 6 '14 at 0:57