G-subsets and G-orbits of Q(sqrt n) under action of the modular group

Abstract

It is well known that $G=⟨x,y:x^2=y^3=1⟩$ represents the modular group $PSL(2,Z)$, where $x:z\to \frac{-1}{z},y:z\to \frac{z-1}{z}$ are linear fractional transformations. Let $n=k^{2}m$, where $k$ is any non zero integer and $m$ is square free positive integer. Then the set $$Q^{\ast }(\sqrt{n}):=\{\frac{a+\sqrt{n}}{c}:a,c,b=\frac{a^2-n}{c}\in Z\text{and}(a,b,c)=1\}$$ is a $G$-subset of the real quadratic field $Q(\sqrt{m})$ \cite{R9}. We denote $\alpha =\frac{a+\sqrt{n}}{c}$ in $Q^{\ast }(\sqrt{n})$ by $\alpha (a,b,c)$. For a fixed integer $s>1$, we say that two elements $\alpha (a,b,c)$, ${\alpha }^{\prime }(a^{\prime },b^{\prime },c^{\prime })$ of $Q^{\ast }(\sqrt{n})$ are $s$-equivalent if and only if $a\equiv a^{\prime }(mods)$, $b\equiv b^{\prime }(mods)$ and $c\equiv c^{\prime }(mods)$. The class $[a,b,c](mods)$ contains all $s$-equivalent elements of $Q^{\ast }(\sqrt{n})$ and $E_s^n$ denotes the set consisting of all such classes of the form $[a,b,c](mods)$. In this paper we investigate proper $G$-subsets and $G$-orbits of the set $Q^{\ast }(\sqrt{n})$ under the action of Modular Group $G$

15 pages and 15 references