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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 $$ \Lambda \backslash SL_2(\mathbb{R}) $$ and $$ \Gamma \backslash SL_2(\mathbb{R})\times SL_2(\mathbb{Q}_p)/K. $$ 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!

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Could you explain what you mean by "isomorphism"? (could you also put parentheses in your double coset? it took me 1 minute to read it: $\Gamma\backslash (\dots\times\dots)/K$). – YCor Jan 18 '13 at 23:49

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