We investigate a particular choice of the Ford fundamental domain of the congruence subgroup <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma _0(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and define a notion of complexity c(N) accordingly, which is a nonnegative integer and carries some information on the shape of the Ford domain. The property that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c(N)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> first appeared as a technical assumption in a paper by Pohl, which is closely related to a conjecture of Zagier on the “reduction theory” of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma _0(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we give a complete classification of positive integers N with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$c(N)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, 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.

The Complexity of Ford Domains of $\Gamma _0(N)$

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2026年4月28日

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Pengcheng Zhang

摘要

We investigate a particular choice of the Ford fundamental domain of the congruence subgroup $\Gamma _0(N)$ $Γ_{0} (N)$ and define a notion of complexity c(N) accordingly, which is a nonnegative integer and carries some information on the shape of the Ford domain. The property that $c(N)=0$ $c (N) = 0$ first appeared as a technical assumption in a paper by Pohl, which is closely related to a conjecture of Zagier on the “reduction theory” of $\Gamma _0(N)$ $Γ_{0} (N)$ . In this paper, we give a complete classification of positive integers N with $c(N)=0$ $c (N) = 0$ , 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.

The Complexity of Ford Domains of \(\Gamma _0(N)\)

摘要