Cusps and fundamental domains for congruence subgroups

TL;DR

The paper addresses the problem of classifying cusp classes and widths for the congruence subgroups $Γ(N)$, $Γ_1(N)$, and $Γ_0(N)$. It develops a uniform combinatorial framework based on $P^1(\mathbb Z/N)$-level data $S(N)$, $S_1(N)$, and $S_0(N)$ and explicit cusp maps $χ$, $χ_1$, $χ_0$, linking cusp structures across subgroups to Nie–Parent's connected fundamental domains via the width-encoding functions $M$ and $W$. Key contributions include explicit bijections between cusp sets and the combinatorial data, closed-form width formulas $wd([a/c])=N$, $wd_1([a/c]_1)=d'$, and $wd_0([a/c]_0)=\tilde{d}$, plus a cusp-width identity for $Γ_0(N)$ expressing widths as a sum over $W$-values; these results provide a computable taxonomy of cusps with compatibility across subgroups. The work advances understanding of how fundamental-domain cusps encode width information, with potential applications to explicit modular-curve computations and automorphic forms.

Abstract

We characterize the cusp classes and their widths for the congruence subgroups $Γ(N), Γ_1(N)$ and $Γ_0(N)$. We relate the cusp classes of $Γ_0(N)$ with those produced by the connected fundamental domain in the previous work of Nie and Parent. By further studying the interesting functions $M$ and $W$ on ${\mathbb Z}/N$, we establish an identity relating the widths.

Theorems & Definitions (21)

• Remark 1.3
• Theorem 1.8
• Proposition 1.15
• Remark 1.25
• Definition 1.26
• Remark 1.28
• Theorem 1.29
• Proposition 1.30
• Theorem 1.35
• proof : Proof of Theorem \ref{['cusps']}