Odd vanishing cycles in cyclotomic fields

TL;DR

This work introduces and analyzes odd vanishing cycles $\Delta^{(1)}_q$ arising from lifts of Hecke group cusps to cyclotomic fields, and develops an odd variant of rank-2 Coxeter-like structures via the odd monodromy group $\Gamma^{(1)}$. It proves that $\Delta^{(1)}_q$ is infinite, discrete, and invariant under $\frac{2\pi}{2q}$-rotations, with explicit minimal unit-point elements and a 2:1 relation to cusp data; it then treats arithmetic cases $q=3,4,6$ explicitly and provides a new constructive proof for $q=5$, showing $G_5(\infty)=\mathbb{Q}(\lambda)\cup\{\infty\}$. The paper further develops the odd Coxeter-like framework by introducing odd monodromy groups, gives presentations for $\Gamma^{(1)}$ and $G_q^{\mathbb C}$ across residue classes of $q$, and surveys lambda-continued fractions and hyperbolic fixed points in this setting. Overall, it connects Hecke groups, cyclotomic fields, and a novel odd- Coxeter-like rank-2 theory, opening avenues toward a systematic theory of odd monodromy data.

Abstract

A cusp of a Hecke group $G_q$ has two natural lifts to the ring of integers of a cyclotomic field. These lifts are called here odd vanishing cycles. All lifts of all cusps together form a discrete subset of ${\mathbb C}$ of some exquisite beauty. They form one or two or four orbits of a certain subgroup of the matrix Hecke group. The subgroup can be considered as a monodromy group and is an analog of a rank 2 Coxeter group, so of a dihedral group. The paper has a research part and a larger survey part.

Figures (12)

• Figure 1.1: Part of the set $\Delta^{(1)}_5$
• Figure 1.2: Part of the set $\Delta^{(1)}_7$
• Figure 4.1: Some hyperbolic lines and triangles
• Figure 4.2: Fundamental domain for $G_q$
• Figure 5.1: The sectors $S(1,\zeta),S(\zeta,\zeta^{q-1})$ and $S(\zeta^{q-1},-1)$

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 2.3
• Corollary 2.4
• Theorem 2.7
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.4
• Definition 3.6
• Lemma 3.7