The complexity of Ford domains of Γ₀(N)

2025/08/25 by Zhang, Pengcheng
#11A99 #11F06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.18511

Abstract

For each integer N>1, we define a notion of complexity c(N), which is a nonnegative integer and carries some information on the shape of the Ford fundamental domain of the congruence subgroup Γ₀(N). The property that c(N)=0 first appeared as a technical assumption in a paper by Anke Pohl, which is closely related to a conjecture of Don Zagier on the "reduction theory" of Γ₀(N). In this paper, we give a complete classification of integers N>1 with c(N)=0, and we also show that c(N) goes to infinity if both the number of distinct prime factors of N and the smallest prime factor of N go to infinity.

Citations

• Reduction theory for Fuchsian groups with cusps
• Strict transfer operator approaches for non-compact hyperbolic orbisurfaces
• Selberg zeta functions, cuspidal accelerations, and existence of strict transfer operator approaches
• Symbolic Dynamics for the Geodesic Flow on Two-dimensional Hyperbolic Good Orbifolds
• Structure of attractors for (a,b)-continued fraction transformations

Related

• Integer Complexity Generalizations in Various Rings
• Théorème de Chebotarev et complexité de Littlewood
• Integer Complexity: Experimental and Analytical Results
• Alternating state complexity of the set of primes and squarefree integers
• Hierarchical Complexity of Finite Groups