Revisiting the action of a subgroup of the modular group on imaginary quadratic number fields

TL;DR

This paper revisits and corrects previous results on how a specific subgroup of the modular group acts on certain subsets of imaginary quadratic fields, providing accurate orbit counts and clarifying earlier errors.

Contribution

The paper identifies and corrects errors in prior work on the action of a subgroup of the modular group on imaginary quadratic fields, offering precise orbit estimates.

Findings

Corrected the estimate for the number of orbits under the subgroup action.

Identified and rectified errors in previous research.

Provided a more accurate understanding of the subgroup's action on quadratic fields.

Abstract

Consider the modular group $\mbox{PSL}(2,\mathbb{Z})=\langle x, \, y \,|\, x^2=y^3=1\rangle$ generated by the transformations $x: z\mapsto -1/z$ and $y:z\mapsto (z-1)/z$. Let $H$ be the proper subgroup $\langle y,\,v\,|\, y^3=v^3=1\rangle$ of $\mbox{PSL}(2,\mathbb{Z})$, where $v=xyx$. The reference (M. Ashiq and Q. Mushtaq, {\em Actions of a subgroup of the modular group on an imaginary quadratic field}, Quasigropus and Related Systems {\bf 14} (2006), 133--146) proposed results concerning the action of $H$ on the subset $\{\frac{a+\sqrt{-n}}{c}\,|\, a,b=\frac{a^2+n}{c}, c \in \mathbb{Z}, c\neq 0\}$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-n})$ for a positive square-free integer $n$. In the current article, the author points out and corrects errors appearing in the aforementioned reference. Most importantly, the corrected estimate for the number of orbits arising from this action is given.