Density of $\Gamma(N)$ in $\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell})$ where $\ell \nmid N$

Question

$\DeclareMathOperator\Sp{Sp}$Let $\Sp_{2g}$ denote the symplectic group of $2g \times 2g$ matrices for some $g \geq 1$, and let $\Gamma(N)$ be the level-$N$ principal congruence subgroup of $\Sp_{2g}(\mathbb{Z})$. It is already known, by certain approximation theorems for algebraic groups, that $\Sp_{2g}(\mathbb{Z})$ is dense in $\Sp_{2g}(\mathbb{Z}_{\ell})$ for any prime $\ell$. Is it known that $\Gamma(N)$ is also dense in $\Sp_{2g}(\mathbb{Z}_{\ell})$ as long as $\ell \nmid N$?

Answer 1

Yes, this also follows from the same approximation theorems.

One very concrete way of stating the strong approximation theorem for $Sp_{2g}$ is that $Sp_{2g}(\mathbf{Z})$ surjects onto $Sp_{2g}(\mathbf{Z} / R \mathbf{Z})$ for every integer $R$. By taking $R = N\ell^k$ for $k \gg 0$, we can find elements of $Sp_{2g}(\mathbf{Z})$ which are congruent to 1 mod $N$ and arbitrarily close $\ell$-adically to anything you like, so $\Gamma(N)$ is dense in $Sp_{2g}(\mathbf{Z}_\ell)$ for any $\ell \nmid N$.