Genus of a subgroup of $\GL_2$ (reviewed)

The genus of an open subgroup $H$ of $\GL_2(\widehat\Z)$ or $\GL_2(\widehat\Z_\ell)$ of level $N$ 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 $\Gamma_H:=\pm H_N \cap \SL_2(\Z/N\Z)$, $H_N$ denotes the projection of $H$ onto $\GL_2(\Z/N\Z)$, and

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

The genus $g(H)$ is a nonnegative integer that equals the genus of each geometric component of the modular curve $X_H$.

This coincides with the genus of the Riemann surface $\mathcal H^*/\Gamma$, where $\mathcal H^*$ is the completed upper half-plane and $\Gamma$ is the inverse image of $\Gamma_H$ in $\SL_2(\Z)$.