Conjugacy classes in PSL(2, K)

TL;DR

This paper explores conjugacy classes in PSL(2, K) over various fields, linking algebraic, geometric, and number-theoretic perspectives, including orbit classification, Pell-Fermat equations, and modular geometry.

Contribution

It provides a comprehensive, field-independent analysis of PSL(2, K) conjugacy classes, connecting algebraic, geometric, and number-theoretic methods in a novel way.

Findings

Classification of orbits for PSL(2, K) actions on sl(2, K)

Partition of PSL(2, Z)-classes of quadratic forms into K-classes

Geometric interpretation of quadratic forms in modular orbifolds

Abstract

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. The synthetic approach enables to change the base field, and we illustrate this picture over the fields with three and five elements, in relation with the geometry of the tetrahedral and icosahedral groups. While the results may appear familiar, they do not seem to be covered in such generality or detail by the existing literature. 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. The results are classical, but our geometrical approach is of independent interest as it may yield new insights into the geometry of Gauss composition, and unify the picture over function fields. 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, in terms of hyperbolic distances and angles between those modular cycles. These geometric quantities are related to linking numbers of modular knots. Their distribution properties could be studied using the geometry of the quadratic lattice (sl(2, Z), det) but such investigations are not pursued here.