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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- 2024-03-25 15:28:32 by Maarten Derickx
- 2024-03-25 15:28:20 by Maarten Derickx
- 2024-03-25 15:27:33 by Maarten Derickx
- 2024-03-22 06:18:04 by Andrew Sutherland
- 2024-03-22 06:11:54 by Andrew Sutherland
- 2022-11-29 07:49:26 by Andrew Sutherland