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For an open subgroup $H$ of $\GL_2(\widehat \Z)$ with $\det(H) =\widehat \Z^\times$, the cusps of $H$ are the double cosets $ \pm H \backslash \GL_2(\widehat \Z) /\langle\delta\rangle$, where $ \delta := \begin{bmatrix}1&1\\0&1\end{bmatrix}$. These correspond to the cusps of the modular curve $X_H$.

Each $\sigma \in \Gal_\Q$ acts on the cusps of $H$ via $Hg\langle \pm \delta\rangle\mapsto Hg\chi(\sigma)\langle \pm \delta\rangle$, where $\chi(\sigma)=\begin{bmatrix}e&0\\0&1\end{bmatrix} \in \GL_2(\widehat \Z)$ is such that for all positive integers $N$ one has $\sigma(\zeta_N)=\zeta_N^e$.

The Galois orbit of a cusp of $H$ is its orbit under the action of $\Gal_\Q$.

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  • Last edited by Maarten Derickx on 2024-03-25 15:28:32
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