Conjugacy classes in ${\mathrm{PSL}}_2\left(𝕂\right)$

Simon, Christopher-Lloyd¹

¹ Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Mathematics Research Reports, Tome 4 (2023), pp. 23-45

Résumé

We first describe, over a field $𝕂$ of characteristic different from $2$, the orbits for the adjoint actions of the Lie groups ${\mathrm{PSL}}_2\left(𝕂\right)$ and ${\mathrm{PSL}}_2\left(𝕂\right)$ on their Lie algebra ${\mathrm{𝔰𝔩}}_2\left(𝕂\right)$. 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 ${\mathrm{PSL}}_2\left(\mathbb{Z}\right)$-classes of integral binary quadratic forms into groups of ${\mathrm{PSL}}_2\left(𝕂\right)$-classes. When $𝕂=ℂ$ we obtain the class groups of a given discriminant. Then we provide a complete description of their partition into ${\mathrm{PSL}}_2\left(\mathbb{Q}\right)$-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 ${\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ for when two points or two closed geodesics correspond to ${\mathrm{PSL}}_2\left(𝕂\right)$-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 $({\mathrm{𝔰𝔩}}_2\left(\mathbb{Z}\right),det)$ but such investigations are not pursued here.

Détail

Reçu le : 2022-09-18
Révisé le : 2023-05-20
Publié le : 2023-12-15

DOI : 10.5802/mrr.16

Classification : 11E04, 22E20, 22E04
Keywords: Lie algebra, adjoint action, killing form, binary quadratic forms, genus class group, Hilbert symbol, modular group, modular cycles

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits