Conjugacy classes of elliptic elements in the Picard group (Q2703135)

The Picard group \(P\) is a non-Fuchsian discrete subgroup of \(\text{PSL}(2,\mathbb{C})\) having Gaussian integer coefficients. \textit{B. Fine} [Can. J. Math. 28, 481-485 (1976; Zbl 0357.20026)] has shown that the total number of conjugacy classes of elliptic elements is seven.NEWLINENEWLINENEWLINEIn this paper, the authors reduce this number to five. They use quadratic forms \(az\overline z+bz+\overline b\overline z+c\) with \(a,c\in\mathbb{Z}\) and \(b\in\mathbb{Z}(i)\). If \(b=b_1+ib_2\), then the discriminant of this form is defined to be \(D=b^2_1+b^2_2-ac\).NEWLINENEWLINENEWLINEClassification is then made according to the discriminant of these forms modulo 4.