Conjugacy classes in PSL(2,K)

Christopher-Lloyd Simon

Published 2022-10-05Version 1

We first describe, over a field K of characteristic different from 2, the orbits for the adjoint actions of the Lie groups PGL(2,K) and PSL(2,K) on their Lie algebra sl(2,K). While the former are well known, the latter lead to the resolution of generalised Pell-Fermat equations which characterise the corresponding orbit. We apply this discussion to partition the set of PSL(2,Z)-classes of integral binary quadratic forms into groups of PSL(2,K)-classes. When K = C we obtain the class groups of a given discriminant. Then we provide a complete description of their partition into PSL(2,Q)-classes in terms of Hilbert symbols, and relate this to the partition into genera. Finally we provide a geometric interpretation in the modular orbifold PSL(2,Z)\H for when two points or two closed geodesics correspond to PSL(2,K)-equivalent quadratic forms.