Connected fundamental domains for congruence subgroups

2024/11/26 by Zhaohu Nie, Nie, Zhaohu, C. Xavier Parent +1
Mathematics · #11F06 #20H05 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2411.17119

openalex publication_date 2024/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We produce canonical sets of right coset representatives for the congruence subgroups Γ₀(N), Γ₁(N) and Γ(N), and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line P¹(\mathbb Z/N\mathbb Z) using a function M: \mathbb Z/N\mathbb Z→ \mathbb Z_{≥ 0}, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function W:\mathbb Z/N\mathbb Z→ \mathbb N, of possible independent interest. We present some examples and pictures at the end.

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