Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\Gamma_n)$ the the center of $\Gamma_n$.

My question is to compute the index $[\Gamma_n/Z(\Gamma_n):\Gamma_{n+1}/Z(\Gamma_{n+1})]$

I know that the index of $[\Gamma_n:\Gamma_{n+1}]=p^4$. I also know that if $G_n$ is the $n$-th congruence subgroup of $SL(2,\mathbb{Z}_p)$, then the index of $[G_n:G_{n+1}]$ is probably $p^3$

(because the topological generators of $G_n$ are $1+p^nE_{2,1}, 1+p^nE_{1,2} $ and $ (1+p^n)E_{1,1}+(1+p^n)^{-1}E_{2,2}$).

Here $E_{i,j}$ is the elementary matrix having $1$ at $(i,j)$-th place and $0$ elsewhere.

I can't figure out the the index $[\Gamma_n/Z(\Gamma_n):\Gamma_{n+1}/Z(\Gamma_{n+1})]$, that is the index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$.

Any help is welcome. Thanks in advance.

A prioriit need not embed in any quotient of $\operatorname{GL}_2(\mathbb Z_p)$.) Also true is that $\operatorname Z(\Gamma_n) \cap \Gamma_{n + 1}$ equals $\operatorname Z(\Gamma_{n + 1})$, so that you're just computing $[\Gamma_n : \Gamma_{n + 1}]\cdot[\operatorname Z(\Gamma_n) : \operatorname Z(\Gamma_{n + 1})]^{-1}$. $\endgroup$ – LSpice Jan 24 at 20:25