On several occasions I heart about the following result:

For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times SL_2(\mathbb{Q}_p)$ and a compact subgroup $K$ of $SL_2(\mathbb{R})\times SL_2(\mathbb{Q}_p)$ such that there is an isomorphism between $$ \Lambda ∖SL_2(\mathbb{R})$$ and $$\Gamma∖SL_2(\mathbb{R})\times SL_2(\mathbb{Q}_p)/K.$$ I know how to prove this for $\Lambda=SL_2(\mathbb{Z})$. Then $\Gamma=SL_2(\mathbb{Z}[1/p])$ (diagonally in $SL_2(\mathbb{R})\times SL_2(\mathbb{Q}_p)$) and $K={1}\times SL_2(\mathbb{Z}_p)$ and the isomorphism is a quite easy map.

I would like to find a reference for more general $\Lambda$, preferably with an explicit statement of the isomorphism and an explanation, what means "certain". Any help is highly appreciated!