Optimal Farey sequence for the Congruence subgroup $Γ_0(2^{n})$

TL;DR

The paper proves that the congruence subgroup $\Gamma_0(2^n)$ ($n\ge2$) admits a special polygon whose Farey denominators $\{e_i\}$ satisfy $e_i\le 2^{n-1}$ for all $i$, with a unique index $j$ where $e_j=2^{n-1}$; each $e_i$ corresponds to a unique cusp parameter $a_i$ yielding independent generators from side pairings. The construction combines an initial Farey block $F_r$ (with $r=\lfloor2^{n/2}\rfloor$) with chains $N_e$ attached to free sides, producing $S_1$, and then augmenting by $a\oplus b$ insertions for $i> n/2$ to obtain the full Farey sequence $T$ for $\Gamma_0(2^n)$; the analysis shows all inserted denominators remain bounded by $2^{n-1}$ and that this bound is optimal. A weaker bound is established for odd primes $p$ via a Fibonacci-type growth control, and an explicit algorithm (DKLT) is provided to compute the Farey sequence for $\Gamma_0(2^n)$. The work extends the geometric understanding of cusps and widths in $\Sigma(N)$, connecting the cusp structure to Farey constructions and providing a practical method for generating independent generators from polygonal fundamental domains. The results offer insight into the arithmetic and geometric properties of congruence subgroups and suggest a general conjecture that the optimal bound should be $p^{n-1}$ for $\Gamma_0(p^n)$ with $n\ge3$.

Abstract

We prove that $Γ_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $Γ_0(2^n)$ whose side-pairings give a set of independent generators of $Γ_0(2^n)$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 5.1
• Proposition 5.2
• Lemma 7.1