Genus formulas for families of modular curves

TL;DR

This paper derives explicit genus formulas for various families of modular curves associated with open subgroups of (\u211d) and provides comprehensive invariants for these formulas.

Contribution

It extends known genus formulas to new families of modular curves, including $X_{ ext{sp}}^+(N)$, $X_{ ext{ns}}^+(N)$, and $X_{ ext{arith},1}(M,MN)$, with explicit invariants and a comprehensive table.

Findings

Explicit genus invariants for new modular curve families.

Complete tables of genus formula invariants provided.

Extension of classical genus formulas to broader modular curve families.

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.