I am working on some first aspects of modular forms and automorphic representations, and would like to understand better the formal dictionary between both. It seems to be essentially based on the following consequence of the strong approximation theorem for adeles, where $G = GL(2)$ and $K$ a compact open subgroup of $G(\widehat{\mathbb{Z}})$ with $\det(K) = \widehat{\mathbb{Z}}$: $$G(\mathbb{Q}) \backslash G(\mathbb{A})/K \simeq \Gamma \backslash G(\mathbb{R})^+,$$

where $\Gamma = G(\mathbb{Q}) \cap G(\mathbb{R})^+ K'$.

Why is this isomorphism true? I guess I should see it through the isomorphism theorem $AH/H \simeq A/A\cap H$, however I do not see in what way write it.


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