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

TL;DR

This paper studies the structure of G-subsets and G-orbits within the set of quadratic irrationals over real quadratic fields, under the action of the modular group, focusing on classifications based on congruence relations.

Contribution

It introduces a detailed analysis of G-subsets and G-orbits of quadratic irrationals in real quadratic fields, emphasizing their classification via modular group actions and congruence classes.

Findings

Classification of G-subsets based on congruence classes.

Description of G-orbits in terms of algebraic and modular properties.

New insights into the structure of quadratic irrationals under modular transformations.

Abstract

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