The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

Abstract

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