Orbits of $\QQ^*(\sqrt{n})$ under the action of Modular Group $PSL(2,\mathbb {Z})$

TL;DR

This paper investigates the orbits of certain quadratic fields under the action of the modular group using coset diagrams and modular arithmetic, providing a detailed classification for various cases involving powers of 2 and odd primes.

Contribution

It introduces a method to determine the G-orbits of specific quadratic fields using coset diagrams and modular arithmetic, extending previous understanding to new cases.

Findings

Classified G-orbits of $bQ^*( oot{p^k}})$ for various $p$ and $k$

Extended orbit analysis to fields involving powers of 2 and odd primes

Provided explicit descriptions of the structure of these orbits

Abstract

Coset diagrams have been used to study quotients, orbits, subgroups and structure of the finitely generated groups. In this paper we use coset diagrams and modular arithmetic to determine the $G$-orbits of $\QQ^*(\sqrt{p^k})$, $\QQ^*(\sqrt{2p^k})$, $\QQ^*(\sqrt{2^2p^k})$, and in general $\QQ^*(\sqrt{2^lp^k})$, for each $l\geq3$ and $k=2h+1\geq3$, for each odd prime $p$.