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I‘m working on an exercise of a book over modular forms (Henri Cohen, Modular Forms, a classical approach) and I’m confused by the frasing of the question: Show that the map $\gamma \mapsto \gamma$i induces a bijection from $SO_2(\mathbb{R})\setminus SL_2(\mathbb{R})$ to $\mathbb{H}$ , where $SO_2(\mathbb{R})$ is the group of rotations. I don‘t know what is ment by $SO_2(\mathbb{R})\setminus SL_2(\mathbb{R})$. I thought of a set difference, but I think this wouldn‘t make sense. In the same book this symbol was also used for a group action, but I don‘t know if this would make sense here. It would be nice if someone could explain the question to me. Thanks a lot, Philipp

Asaf Karagila
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  • In fact it is $SL_2(\mathbb{R})/SO_2(\mathbb{R})$ the quotient of a group by a subgroup, then $SL_2(\mathbb{R})$ acts on the left on these cosets and $\gamma SO_2(\mathbb{R}) \to \gamma i$ is an homeomorphism compatible with the action. – reuns May 14 '21 at 12:27
  • $ SL_2(\mathbb{Z})$ left-acts on $\Bbb{H}$ and the quotient space ${ SL_2(\mathbb{Z})z,\Im(z)>0}$ is written $SL_2(\mathbb{Z})\backslash \Bbb{H}$. – reuns May 14 '21 at 12:28
  • I have also been confused many times by this. Apparently there is a distinction between "/" and "" when taking quotients. One denotes left actions and the other denotes right actions. But not everybody seems to follow this convention. – Giuseppe Negro May 14 '21 at 12:53

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