Coset diagram for the action of Picard group on \(\mathbb{Q}(i,\sqrt{3})\) (Q2796959)

Let \(\mathbb{Z}[i]\) be the ring of Gaussian integers. The Picard group \(\Gamma =\mathrm{PSL}(2,\mathbb{Z}[i])\) is the group of linear fractional transformations NEWLINE\[NEWLINET(z)=\frac{az+b}{cz+d},NEWLINE\]NEWLINE where \(ad-bc=1\) and \(a,b,c,d\in \mathbb{Z}[i]\). In the paper under review, the authors deal with the behavior of elements as words in orbits of the action of the Picard group on \(\mathbb{Q}[i,\sqrt{3}]\) using coset diagrams. The behavior of ambiguous numbers in the bi-quadratic field \(\mathbb{Q}[i,\sqrt{3}]\) is investigated by the action of the Picard group on it. They also give a procedure to obtain ambiguous numbers of the form \(\frac{a+b\sqrt{3}}{c}\) where \(b\) is a positive integer.