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

zbMATH Open1291.20050arXiv1009.4619MaRDI QIDQ5168819

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Publication date: 21 July 2014

Abstract: It is well known that $G=langlex,y:x^2=y^3=1angle$ represents the modular group $PSL(2,Z)$, where $x:zightarrowfrac-1z,y:zightarrowfracz-1z$ 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 ${\displaystyle }$Q^*(sqrt{n}):={frac{a+sqrt{n}}{c}:a,c,b=frac{a^2-n}{c}in Z~ extmd{and}~(a,b,c)=1}${\displaystyle }$ is a $G$-subset of the real quadratic field $Q(sqrtm)$ cite{R9}. We denote $alpha=fraca+sqrtnc$ in $Q^{\ast }(sqrtn)$ by $alpha(a,b,c)$. For a fixed integer $s>1$, we say that two elements $alpha(a,b,c)$, $alph{a}^{\prime }(a^{\prime },b^{\prime },c^{\prime })$ of $Q^{\ast }(sqrtn)$ are $s$-equivalent if and only if $aequiv{a}^{\prime }(mods)$, $bequiv{b}^{\prime }(mods)$ and $cequiv{c}^{\prime }(mods)$. The class $[a,b,c](mods)$ contains all $s$-equivalent elements of $Q^{\ast }(sqrtn)$ 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 $Q^{\ast }(sqrtn)$ under the action of Modular Group $G$

Full work available at URL: https://arxiv.org/abs/1009.4619

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