Genus formulas for families of modular curves

Abstract

For each open subgroup $H\le {\mathrm{GL}}_2(\hat{\mathbb{Z}})$, there is a modular curve $X_H$, defined as a quotient of the full modular curve $X(N)$, where $N$ is the level of $H$. The genus formula of a modular curve is well known for $X_0(N)$, $X_1(N)$, $X(N)$, $X_{\mathrm{sp}}(N)$, $X_{\mathrm{ns}}(N)$, and $X_{S_4}(p)$ for $p$ prime. We explicitly work out the invariants of the genus formulas for $X_{\mathrm{sp}}^{+}(N)$, $X_{\mathrm{ns}}^{+}(N)$, and $X_{\text{arith},1}(M,MN)$. In Table $1$, we provide the invariants of the genus formulas for all of the modular curves listed.