The Hecke group $H({\lambda }_4)$ acting on imaginary quadratic number fields

Abstract

Let $H({\lambda }_4)$ be the Hecke group $⟨x,y:x^2=y^4=1⟩$ and, for a square-free positive integer $n$, consider the subset ${\mathbb{Q}}^{\ast }(\sqrt{-n})=\left\{(a+\sqrt{-n})/c|a,b=(a^2+n)/c\in \mathbb{Z},c\in 2\mathbb{Z}\right\}$ of the quadratic imaginary number field $\mathbb{Q}(\sqrt{-n})$. Following a line of research in the relevant literature, we study properties of the action of $H({\lambda }_4)$ on ${\mathbb{Q}}^{\ast }(\sqrt{-n})$. In particular, we calculate the number of orbits arising from this action for every such $n$. Some illustrative examples are also given.