Genus formulas for families of modular curves
Asimina S. Hamakiotes, Jun Bo Lau
TL;DR
The work addresses computing explicit genus formulas for families of modular curves X_H attached to open subgroups of GL_2(̂Z), extending known results to new Cartan-type families and arithmetic covers. It develops a framework based on the PSL_2-index, elliptic points, cusps, and the Riemann-Hurwitz formula, augmented by local prime-power analyses for split and non-split Cartan subgroups and their normalizers. The authors provide closed-form invariants for X_sp^+(N), X_ns^+(N), and X_arith,1(M,MN), summarize them in Table 1, and connect these results to LMFDB entries, enabling direct arithmetic investigations. The methods yield multiplicative behavior across levels and deliver practical tools for studying rational points and potential uniformity questions in Galois representations tied to elliptic curves.
Abstract
For each open subgroup $H\leq \operatorname{GL}_2(\widehat{\mathbb{Z}})$, 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.