The Hecke Group Hλ4 Acting on Imaginary Quadratic Number Fields

Abstract

Let Hλ4 be the Hecke group x,y:x2=y4=1 and, for a square-free positive integer n, consider the subset ℚ∗−n=a+−n/c|a,b=a2+n/c∈ℤ, c∈2ℤ of the quadratic imaginary number field ℚ−n. Following a line of research in the relevant literature, we study the properties of the action of Hλ4 on ℚ∗−n. In particular, we calculate the number of orbits arising from this action for every such n. Some illustrative examples are also given.