Coset Diagram for the Action of Picard Group on Q(i,\surd3)

TL;DR

This paper studies the action of the Picard group on the biquadratic field Q(i,√3), focusing on ambiguous numbers, their patterns, and how to generate them, revealing finite closed paths in their orbits.

Contribution

It introduces a coset diagram for the Picard group's action on Q(i,√3), characterizes ambiguous numbers, and provides a method to generate and analyze their finite orbits.

Findings

Ambiguous numbers form a unique pattern in the coset diagram.

Finite number of ambiguous numbers exist in each orbit, forming a closed path.

A procedure to generate ambiguous numbers of a specific form is devised.

Abstract

The Picard group {\Gamma} is PSL(2,Z[i]). We have defined coset diagram for the Picard group. It has been observed that some elements of Q(i,/surd3) of the form ((a+b/surd3)/c) and their conjugates ((a-b/surd3)/c) over \mathbb{Q}(i) have different signs in the coset diagram for the action of {\Gamma} on the biquadratic field Q(i,/surd3), these are called ambiguous numbers. We have noticed that ambiguous numbers in the coset diagram for the action of {\Gamma} on \mathbb{Q}(i,/surd3) form a unique pattern. It has been shown that there are finite number of ambiguous numbers in an orbit {\Gamma}{\alpha}, where {\alpha} is ambiguous, and they form a closed path and it is the only closed path in the orbit {\Gamma}{\alpha}. We have devised a procedure to obtain ambiguous numbers of the form ((a+k/surd3)/c), where k is a positive integer.