Conjugacy classes in PSL 2 (𝕂)
Mathematics Research Reports, Volume 4 (2023), pp. 23-45.

We first describe, over a field 𝕂 of characteristic different from 2, the orbits for the adjoint actions of the Lie groups PSL 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 ()-classes of integral binary quadratic forms into groups of PSL 2 (𝕂)-classes. When 𝕂= we obtain the class groups of a given discriminant. Then we provide a complete description of their partition into PSL 2 ()-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 () 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 (),det) but such investigations are not pursued here.

Published online:
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
Christopher-Lloyd Simon 1

1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Christopher-Lloyd Simon},
     title = {Conjugacy classes in $\mathrm{PSL}_2(\mathbb{K})$},
     journal = {Mathematics Research Reports},
     pages = {23--45},
     publisher = {MathOA foundation},
     volume = {4},
     year = {2023},
     doi = {10.5802/mrr.16},
     language = {en},
     url = {}
AU  - Christopher-Lloyd Simon
TI  - Conjugacy classes in $\mathrm{PSL}_2(\mathbb{K})$
JO  - Mathematics Research Reports
PY  - 2023
SP  - 23
EP  - 45
VL  - 4
PB  - MathOA foundation
UR  -
DO  - 10.5802/mrr.16
LA  - en
ID  - MRR_2023__4__23_0
ER  - 
%0 Journal Article
%A Christopher-Lloyd Simon
%T Conjugacy classes in $\mathrm{PSL}_2(\mathbb{K})$
%J Mathematics Research Reports
%D 2023
%P 23-45
%V 4
%I MathOA foundation
%R 10.5802/mrr.16
%G en
%F MRR_2023__4__23_0
Christopher-Lloyd Simon. Conjugacy classes in $\mathrm{PSL}_2(\mathbb{K})$. Mathematics Research Reports, Volume 4 (2023), pp. 23-45. doi : 10.5802/mrr.16.

[1] V. Arnold Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane, J. Geom. Phys., Volume 53 (2005) no. 4, pp. 421-427 | DOI | MR

[2] Élie Cartan Leçons sur la géométrie projective complexe. La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile. Leçons sur la théorie des espaces à connexion projective, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1992, vii+911 pages (Reprint of the editions of 1931, 1937 and 1937) | MR

[3] J. W. S. Cassels Rational quadratic forms, London Mathematical Society Monographs, 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978, xvi+413 pages | MR

[4] John Conway; Francis Fung The sensual (quadratic) form, MAA, 1997

[5] David Cox Primes of the form x 2 +ny 2 , Wiley-Interscience, 1997

[6] Jean Dieudonné La géometrie des groupes classiques, Springer-Verlag, 1971

[7] Carl Friedrich Gauss Recherches Arithmétiques, Courcier, 1807

[8] Étienne Ghys; Jos Leys Lorenz and modular flows: a visual introduction, 2016 (

[9] Nikolai Ivanov Arnol’d, the Jacobi identity, and orthocenters, Amer. Math. Monthly, Volume 118 (2011) no. 1, pp. 41-65 | DOI | MR | Zbl

[10] Robert Penner The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys., Volume 113 (1987) no. 2, pp. 299-339 | DOI | MR | Zbl

[11] Robert Penner The geometry of the Gauss product, J. Math. Sci., Volume 81 (1996) no. 3, pp. 2700-2718 | DOI | MR | Zbl

[12] Jean-Pierre Serre Cours d’arithmétique, PUF, 1970

[13] Christopher-Lloyd Simon Arithmetic and Topology of Modular knots, Thèse, Université de Lille, June (2022) (PDF on HAL)

[15] André Weil Number theory: An approach through history From Hammurapi to Legendre, Birkhäuser, 1984, xxi+375 pages | DOI | MR

Cited by Sources: