A note on the action of Hecke groups on subsets of quadratic fields

Abstract

We study the action of the groups $H(\lambda )$ generated by the linear fractional transformations $x:z\mapsto -\frac{1}{z}\text{ and }w:z\mapsto z+\lambda$, where $\lambda$ is a positive integer, on the subsets ${\mathbb{Q}}^{\ast }(\sqrt{n})=\{\frac{a+\sqrt{n}}{c}|a,b=\frac{a^2-n}{c},c\in \mathbb{Z}\}$, where $n$ is a square-free integer. We prove that this action has a finite number of orbits if and only if $\lambda =1$ or $\lambda =2$, and we give an upper bound for the number of orbits for $\lambda =2$.

arXiv comment6 pages