I have two questions related to congruence subgroups.
Let $\Gamma=\Gamma_0(N)=\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \subset SL_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \}$ be a congruence subgroup.
I am wondering why $\Gamma \backslash SL_2(\mathbb{R}) \simeq Z(\mathbb{R})\Gamma' \backslash GL_2(\mathbb{R})$, where $\Gamma'=\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \subset GL_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \}$. I think it might be true but I cannot prove it rigorously.
For $\alpha \in GL_2(\mathbb{Q})$, consider the double coset decomposition $\Gamma \alpha \Gamma = \bigcup \Gamma a_i$ in $GL_2(\mathbb{Q})$, where $a_i \in GL_2(\mathbb{Q})$. Let $\mathbb{A}_f$ be the finite adele ring of $\mathbb{Q}$ and let $U_{\Gamma} \subset GL_2(\mathbb{A}_{f})$ be an open compact subgroup such that $U_{\Gamma} \cap GL_2(\mathbb{Q})=\Gamma.$ I am wondering why the double coset decomposition $U_{\Gamma}\alpha U_{\Gamma}=\bigcup U_{\Gamma}a_i$ holds in $GL_2(\mathbb{A}_{f})$.
Very thanks in advance!