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 $X_H$.
The width of a cusp $c \in \pm H \backslash \GL_2(\widehat \Z) /\langle \delta\rangle$ is the number of right cosets it contains, or equivalently the smallest integer $h$ such that $c\delta^h = c$. The sum of the cusp widths is equal to the $\PSL_2$-index of $H$.
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- Last edited by Maarten Derickx on 2024-03-22 06:32:00
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- 2024-03-22 06:32:00 by Maarten Derickx
- 2024-03-22 06:31:45 by Maarten Derickx
- 2024-03-22 06:18:49 by Andrew Sutherland
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- 2024-03-21 10:18:59 by Maarten Derickx
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- 2022-12-12 07:15:04 by Andrew Sutherland
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