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The genus of an open subgroup $H$ of $\GL(2,\widehat\Z)$ or $\GL(2,\widehat\Z_\ell)$ is defined by the formula \[ g(H) = g(\Gamma_H) = 1+\frac{i(\Gamma_H)}{12} - \frac{\nu_2(\Gamma_H)}{4} - \frac{\nu_3(\Gamma_H)}{3} - \frac{\nu_\infty(\Gamma_H)}{2}, \] where $\Gamma_H$ is the congruence subgroup $\pm H\cap \SL_2(N)$, where $N$ is the level of $H$, and

  • $i(\Gamma_H)=[\SL_2(N):\Gamma_H]$,
  • $\nu_2$ is the number of right cosets of $\Gamma_H$ in $\SL_2(N)$ that contain $\begin{bmatrix}0&1\\-1&0\end{bmatrix}$,
  • $\nu_3$ is the number of right cosets of $\Gamma_H$ in $\SL_2(N)$ that contain the matrix $\begin{bmatrix}0&1\\-1&-1\end{bmatrix}$,
  • $\nu_{\infty}(\Gamma_H)$ is the number of orbits of the right coset space $\Gamma_H\backslash \SL_2(N)$ under the right action of $\begin{bmatrix}1&1\\0&1\end{bmatrix}$.

The genus $g(H)$ is a nonnegative integer that is equal to the genus of each of the geometrically connected components of the modular curve $X_H$.

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  • Review status: beta
  • Last edited by Andrew Sutherland on 2021-07-17 11:56:34
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